# Full text of "An elementary treatise on quaternions"

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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ Boiand NOV 9 1899 SCIENCE CENTER LIBRARY ..(yvL..- .£..- ...,.../.^.<^.f. Clacma^mt ||»8s §^mt» QUATERNIONS TAIT %oxition MACMILLAN AND CO. PUBLISHERS TO THE UNIVERSITY OP o AN ELEMENTARY TREATISE ON aUATERNIONS P. O. TAIT, M.A. rORMBRLY rSLLOW OF ST. PSTSR'S COLLBOR, CAMBRIDOR, PROVRSSOR OP NATURAL PHILOSOPHY ZH THR UNIVRRSITY OP ROIHBUROH. warfdof dfydotf <pviX€OM fiii&ffuiT* ixowraaf. AT THE CLARENDON PRESS ILDCCCiXVn. [The righti af tirantlaUon and reproduction are reserved,] Moih lUiXi. /0,J.V-cVu^''f S'' c<^. ^ <:w^:v/wv:< PEEFACE. The present work was commenced in 1859^ while I was a Professor of Mathematics^ and far more ready at Quaternion analysis than I can now pretend to be. Had it been then completed I should have had means of testing its teaching capabilities^ and of improving it^ before publication^ where found deficient in that respect. The duties of another Chair^ and Sir W. Hamilton's wish that my volume should not appear till after the publication of his Elements J interrupted my already extensive preparations. I had worked out nearly all the examples of Analytical Geometry in Todhimter's Collection^ and I had made various physical appli- cations of the Calculus^ especially to Crystallography^ to Geo- metrical Optics, and to the Induction of Currents, in addition to those on Kinematics, Electrodynamics, &c., which are re- printed in the present work from the Quarterly Mathematical Jov/mal and the Proceedmgs of the Royal Society of Edinburgh. Sir W. Hamilton, a few days before his death, urged me to prepare my work as soon as possible, his being almost ready for publication. He then expressed, more strongly perhaps than he had ever done before, his profound conviction of the impor- tance of Quaternions to the progress of physical science ; and VI PREFACE. his desire that a really elementary treatise on the subject should soon be published. I regret that I have so imperfectly fulfilled this last request of my revered Mend. When it was made I was already engaged, along with Sir W. Thomson, in the laborious work of preparing a large Treatise on Natural Philosophy. The present volume has thus been written under very disadvantageous circumstances, especially as I have not found time to work up the mass of materials which I had originally collected for it, but which I had not put into a fit state for publication. I hope, however, that I have to some extent succeeded in producing a thoroughly elementary work> intelligible to any ordinary student ; and that the numerous examples I have given, though not specially chosen so as to display the full merits of Quaternions, will yet sufficiently show their admirable simplicity and naturalness to induce the reader to attack the Lectures and the Elements; where he will find, in provision, stores of valuable results, and of elegant yet powerM analytical investigations, such as are contained in the writings of but a very few of the greatest mathematicians. For a succinct account of the steps by which Hamilton was led to the invention of Quaternions, and for other interesting information regarding that remarkable genius, I may refer to a slight sketch of his life and works in the North British Review for September 1866. It will be found that I have not servilely followed even so great a master, although dealing with a subject which is en- tirely his own. I cannot, of course, tell in every case what I have gathered from his published papers, or from his voluminous correspondence, and what I may have made out for myself. Some theorems and processes which I have given, though wholly my own, in the sense of having been made out for myself before the publication of the Memetits, I have since found there. Others PREFACE. vii also may be, for I have not yet read that tremendous volume completely, since much of it bears on developments unconnected with Hiysics. But I have endeavoured throughout to point out to the reader all the more important parts of the work which I know to be wholly due to Hamilton. A great part, indeed, may be said to be obvious to any one who has mastered the preliminaries j still I think that, in the two last Chapters especially, a good deal of orig^al matter will be found. The volume is essentially a working one, and, especially in the later Chapters, is rather a collection of examples than a detailed treatise on a mathematical method. I have constantly aimed at avoiding too great extension; and in pursuance of this object have omitted many valuable elementary portions of the subject. One of these, the treatment of Quaternion logarithms and exponentials, I greatly regret not having given. But if I had printed all that seemed to me of use or interest ^ to the student, I might easily have rivalled the bulk of one of Hamilton's volumes. The beginner is reconmiended merely to read the first five Chapters, then to work at Chapters VI, VII, VIII (to which numerous easy Examples are appended). After this he may work at the first five, with their (more difficult) Examples ; and the remainder of the book should then present no difficulty. Keeping always in view, as the great end of every mathe- matical method, the physical applications, I have endeavoured to treat the subject as much as possible from a geometrical instead of an analytical point of view. Of course, if we premise the properties of i,/, k merely, it is possible to construct from them the whole system^; just as we deal with the imaginary of * This has been done by Hamilton himself, as one among many methods he has employed ; and it is also the foundation of a memoir by M. All^gret entitled Eisai 8ur le Galcul des Qttatemions (Paris, 1863). Vlli PBBFACE. Algebra, or, to take a closer analogy, just as Hamilton himself dealt with Couples, Triads, and Sets. This may be interesting to the pure analyst, but it is repulsive to the physical student, who should be led to look upon i,y, h from the very first as geometric realities, not as algebraic imaginaries. The most striking peculiarity of the Calculus is that muliipli- cation is not generally commutative^ i. e. that qr is in general different from rq, r and q being quaternions. StiU it is to be remarked that something similar is true, in the ordinary coor- dinate methods, of operators and functions : and therefore the student is not wholly unprepared to meet it. No one is puzzled by the fact that log. cos. ^ is not equal to cos.log.a?, or that ^/j- is not equal to ^ Vy. Sometimes, indeed, this rule is most absurdly violated, for it is usual to take cos' a? as equal to (cos a?)», « while cos"*^ is not equal to (cosa?)"'^ No such incongruities appear in Quaternions ; but what is true of operators and frmctions in other methods, that they are not generally com- mutative, is in Quaternions true in the multiplication of (vector) coordinates. It will be observed by those who are acquainted with the Calculus that I have, in many cases, not given the shortest or simplest proof of an important proposition. This has been done with the view of including, in moderate compass, as great a variety of methods as possible. With the same object I have endeavoured to supply, by means of the Examples appended to each Chapter, hints (which will not be lost to the intelligent student) of further developments of the Calculus. Many of these are due to Hamilton, who, in spite of his great originality, was one of the most excellent examiners any University can boast of. It must dways be remembered that Cartesian methods are PREFACE. IX mere particular cases of Quaternions, where most of the dis- tinctive features have disappeared; and that when, in the treat- ment of any particular question, scalars have to be adopted^ the Quaternion solution becomes identical with the Cartesian one. Nothing therefore is ever lost, though much is generally gained, by employing Quaternions in preference to ordinary methods. In fact, even when Quaternions degrade to scalars, they give the solution of the most general statement of the problem they are applied to, quite independent of any limitations as to choice of particular coordinate axes. There is one very desirable object which such a work as this may possibly fulfil. The University of Cambridge, while seeking to supply a real want (the deficiency of subjects of examination for mathematical honours, and the consequent fre- quent introduction of the wildest extravagance in the shape of data for " Problems^'), is in danger of making too much of such elegant trifles as Trilinear Coordinates, while gigantic systems like Invariants (which, by the way, are as easily introduced into Quaternions as into Cartesian methods) are quite beyond the amount of mathematics which even the best students can master in three years' reading. One grand step to the supply of this want is, of course, the introduction into the scheme of examina- tion of such branches of mathematical physics as the Theories of Heat and Electricity. But it appears to me that the study of a mathematical method like Quaternions, which, while of immense power and comprehensiveness, is of extraordinary simplicity, and yet requires constant thought in its applications, would also be of great benefit. With it there can be no " shut your eyes, and write down your equations/' for mere mechanical dexterity of analysis is certain to lead at once to error on account of the novelty of the processes employed. The Table of Contents has been drawn up so as to give the b X PREFACE. student a short and simple smnmary of the chief fundamental formulae of the Calculus itself^ and is therefore confined to an analysis of the first five chapters. In conclusion^ I have only to say that I shall be much obliged to any one^ student or teacher^ who will point out portions of the work where a difficulty has been found ; along with any inaccuracies which maybe detected. As I have had no assistance in the revision of the proof-sheets^ and have composed the work at irr^;ular intervals^ and while otherwise laboriously occupied^ I fear it may contain many slips and even errors. Should it reach another edition there is no doubt that it will be improved in many important particulars. P. G. TAIT. COLLBGS, EblHBUBGH, Jvly 1867. CONTENTS. Chapter I. — ^Vectors and their Composition 1-29 Sketch of the attempts made to represent geometrically the imaginary of algebra, §§ 1-13. De Moivre's Theorem interpreted in plane rotation, § 8. Gniious speoala- tion of Servois, § 11. Elementary geometrical ideas connected with relative position, § 15. De- finition of a Veotob. It may be employed to denote transloHan, §16. Expression of a vector by one symbol, containing implicitly three distinct numbers. Extension of the signification of the symbol «, § 18. The sign + defined in accordance with the interpretation of a vector as representing translation, § 19. Definition of — . It simply reverses a vector, § 20. Triangles and polygons of vectors, analogous to those of forces and of simnltaneons velocities, § 21. When two vectors are paraUd we have ' a»2^, § 22. Any vector whatever may be expressed in terms of three distinct vectors, which are not coplanar, by the formula p^xa + yfi + zy, which exhibits the three numbers on which the vector depends, § 28. Any vector in the same plane with a and fi may be written p^xa + yfi, § 24. The equation •■=Pf between two vectors, is equivalent to three distinct equations among numbers, § 25. xii CONTENTS. The Commutative and Associative Laws hold in the combination of vectors by the signs + and — , § 27. The equation p = x$, where p is a variable, and fi a fixed, vector, represents a line drawn through the origin parallel to 0. p=^a + x0 is the equation of a line drawn through the extremity of a and parallel to /B, § 28. p = ya + x$ represents the plane through the origin parallel to a and fi, § 29. The condition that p, a, fi may terminate in the same line is pp + qa + rfi = 0, subject to the identical relation p + q + r = 0. Similarly pp + qa + rfi + 8y==0, with, p + q + r + 8 = 0y is the condition that the extremities of four vectors lie in one plane, § 30. Examples with solutions, § 31. Differentiation of a vector, when given as a function of one number, §§ 32-38. If the equation of a curve be P = it>{8) where a is the length of the arc, dp is a vector tangent to the curve, and its length is ds, §§ 38, 39. Examples with solutions, §§ 40-44. Examples to Chapter 1 29-81 Chapter II. — Products and Quotients of Vectors ... 32-59 Here we begin to see what a quaternion is. When two vectors are parallel their quotient is a number, §§ 45, 46. When they are not parallel the quotient in general involves faur distinct numbers — and is thus a Quaternion, § 47. A quaternion regarded as the operator which turns onft vector into another. It is thus decomposable into two factors,' whose order is indifferent, the stretching factor or Tensor, and the turning factor or Versor. These are denoted by Tq and Uq, § 48. The equation fi = qa gives — — 3» or i8<»~^ = g'» hut not in general a-^fi^q, § 49. q or $a-^ depends only on the relative lengths, and directions, of fi and a, §50. CONTENTS. xiu Reciprocal of a quaternion defined, 5 = - gives- or q » = -, T.?-^=^-, ILq-'^-^Uq, §51. Definition of the Conjiigate of a quaternion, Kq^{Tqfq-\ and qKq = Kq.q = {Tq)\ §52. Representation of yersors by arcs on the unit-sphere, § 53. Versor multiplication illustrated by the composition of arcs, § 54. Proof that K{qr) = Kr.Kq, §55. Proof of the Associative Law of Multiplication p.qr=^pq.r, §§ 57-60. [Digression on Spherical Conies, § 59*.] Quaternion addition and subtraction are commutative, § 61. Quaternion multiplication and division are distribtUivet § 62. Composition of quadrantal versors in planes at right angles to each other. Calling them t, j, k, we have i»=ja=&«=_l. ij^-^ji=h, jk=-ij=i, ki=^-ik^j, ijk=-l, §§ 64-71. A unit-vector, when employed as a factor, may be considered as a qua- drantal versor whose plane ia perpendicular to the vector. Hence the equations just written are true of any set of rectangular unit-vectors i,j,k, §72. The product, and quotient, of two vectors at right angles to each other is a third perpendicular to both. Hence J5Ca=-.a, and (T'a)«=aKi«=-o», § 78. Eveiy versor may be expressed as a power of some unit- vector, § 74. Every quaternion may be expressed as a power of a vector, § 76. The Index Law is true of quaternion multiplication and division, § 76. Quaternion considered as the sum of a Soalab and Yectob. q::^-^x + y^Sq+Vq, §77. a Proof that SKq^^Sq, VKq=^ - Vq, § 79. Quadrinomial expression for a quaternion q=^w + ix+jy + kz. An equation between quaternions is equivalent to four equations between numbers (or scalars), § 80. Second proof of the distributive law of multiplication, § 81. Algebraic determination of the constituents of the product and quotient of two vectors, §§ 82-84. XIV C0> TENTS. Heewj^ prryiif o^ the nM9fi0'yttiz*. Uw of mTiItli>L:«:aar»n, § *5. PrrjT/f of the 6>nQ'ilAe Proof of the iomnilae Fu^^- F.7a«, = Faa9ya+ F^&i5+ F7«5A §$ 90.92. Hamilton'f proof thai the product of two pMallel Tecton must be a scalar, and thai of perpendicalar vecton, a vector ; if qvatanions are to deal with space indifferently in all directions, § 93. Examples to Chapter II 59, 60 Chapteb ni. — Intekpeetations and Transpormattons op QUATEBNION EXPRESSIONS 61-85 If be the angle between two yectors, a and 0, we have 5^- = ^^c08«, Sa0=^TaT$cos$, TV-^^sm0, TVafi=TaTfiam$. a la Applications to plane trigonometiy, §§ 94-97. Safi^O shows that a is perpendicular to /3, while Va$=0, shows that a and $ are paralleL 8,a0y is the Yolmne of the parallelepiped three of whose contenninouB edges are a, fi,y. Hence 5.0^7=0 shows that a, $,y are coplanar. Expression of S.affy as a determinant, §§ 98-102. Proof that (TqY^iSqy^ + {TVq)\ and T(2r) = rgrr, §108. Simple propositions in plane trigonometry, § 104. Proof that ofiaT^ is the vector reflected ray, when /9 is the incident ray and a normal to the reflecting surface, § 105. CONTENTS. XV Interpretation of oiB^ when it is a vector, § 106. ' Examples of variety in simple transformations, § 107. Introduction to spherical trigonometry, §§ 108-113. Representation, graphic, and by quaternions, of the spherical eicess, §§ 114, 116. Lod represented by different equations — ^points, lines, surfaces, and solids, §§ 116-119. Proof that ^ r-»(r«2^*5-i«i7(r2 + J&iC2). § 120. Proof of the transformation BiQUATESNIONS, §§ 123-125. Conyenient abbreviations of notation, §§ 126, 127. £cXAMFLfls TO Chafteb III 85-8S Chaptbe rV. — Differentiation of Quatebnions. . . 89-96 Definition of a differential, dr^dFi^^n j i? (j + ^-) -Fq\, where dq is any quaternion whatever. We may write dFq=^f{q,dq), where / is linear and homogeneous in dq ; but we cannot generally write dFq=f(q)dq, §§128-131. Definition of the differential of a function of more quaternions than one. d (qr) =s qdr + dq.r, but not generaUy d (qr) «» qdr + rdq. § 132. Proof that dTp ^dp ^-r^,&c..§133. Up p Successive differentiation; Taylor's theorem, §§ 134, 135. If the equation of a surface be the differential may be written Svdp=^0, where v is a vector normal to the sur&ce, § 137. ExAHPLBS TO Chapter IY 96 XVI CONTENTS. Chaptee V. — ^Thb Solution of Equations of the Fibst Degree. 97-128 The most general equation of the first degree in an unknown quaternion q, may be written ^VMqh-\ 8.cq = d, where a, h, c, d 9ie given quaternions. Elimination of Sq, and reduc- % tion to the vector equation ^p = 2.oSi3p = 7, §§ 138, 139. General proof that ^'p is expressible as a linear function oi p, <pp, and ^'''p, § 140. Value of ^ for an ellipsoid, employed to illustrate the general case, §§ 141-143. Hamiltop's solution of If we write Sa^ = Sp^'tr^ the functions ^ and ^' are said to be conjugate, and Proof that m, whose value may be written as is the same for all values of A, ijl^v, §§ 144-146. Proof that if mg^m-\rm^g-\-m^g^ +g^, where and S {\fJUt>'v + p'XfJLP + A^ V) S.Kfiv * then mg{<p + g)~^ V\fi = (m<f>~^ + g'X + </'*) ^^M- Also that X = »*a — ^> whence the final form of solution m<p-^ =:mi -m,^ + ^% §§ 147, 148. Examples, §§ 149-161. The fundamental cubic ip'^-m^<t>^ + m^il>-m'='{<p-g^){<f,-g^){<p-g^) = 0. When ^ is its own conjugate, the roots of the cubic are real; and the equation yp^P = ^t or {<t>-g)p=0, is satisfied by a set of three real and mutually perpendicular vectors. Geometrical interpretation of these results, §§ 162-166. ^ Proof of the transformation <t>p ^pp + q V.i% + €k)p{l — ck) where (<^ — </i)* = 0, {<l>-g,)k = 0, CONTENTS. xvii 9x-9% P^i(9i+9t\ ^^-i(9i-9i)' Another transformation is il>pm,aaVap + h$S$p, §§ 167-169. Other properties of ^. Proof that Sp(!p+9)''^P^0, and Spdp + h)'^ p^'^O represent the same snrfiice if Proof that when ^ is not self-conjugate 4^^4/p+V€p. Proof that, if g^o^ + iS^/S + T^, where a, iS, 7 are any rectangular unit vectors whatever, we have Sq^-m^, Fg«€. §§170-174. Degrees of indeterminateness of the solution of a quaternion equation — Examples, §§ 175-179. The linear function of a quaternion is given by a symbolical biquadratic, §180. Particular forms of linear equations, §§ 181-183. A quaternion equation of the mth degree in general involves a scalar equation of degree mS § 184. Solution of the equation q*^qa + h, § 185. EXAMFLBS TO Chafteb V 128-130 Chaptbk VI. — Oeometbt op the Straight Line and Plane. 131-147 ExAMPT.iBH TO Chaiteb YL 148-150 Chapter VII. — ^Thb Sphere and Cyclic Cone. . . 151-167 EXAUFLBS TO Chapteb YII 167-170 Chapter VIII. — Surfaces op the Second Order. . 171-191 Examples to Chaftib Yin 192-ld6 C xviu CONTENTS. Chapter IX. — Geometry of Curves and Surfaces. 197-238 Examples to Chaftbr IX 238-247 Chapter X. — Kinematics . , . 248-271 Examples to Chapter X 271-275 Chapter XI. — ^Physical Applications 276-311 Miscellaneous Examples 312-320 ERRATA. Page 171, first line of § 249, fw OD read OD. ,, 213, last line but one, for Sctfip read S.afip. ,, 225, Une 10, /or equations read equation. QUATERNIONS. F' CHAPTER I. VECTORS, AND THEIR COMPOSITION. ^OR more than a century and a half the geometrical representation of the negative and imaginary alge- braic quantities,— 1 and \/— 1, or, as some prefer to write them, — and — ^, has been a favourite subject of speculation with mathematicians. The essence of almost all of the proposed processes consists in employing such quantities to indibate the direction, not the length, of lines. 2. Thus it was soon seen that if positive quantities were measured off in one direction along a fixed line, a useful and lawful convention enabled us to express negative quantities by simply laying them off on the same line in the opposite direction. This convention is an essential part of the Cartesian method, and is constantly employed in Analytical Geometry and Applied Mathematics. 3. Wallis, in the end of the seventeenth century, proposed to represent the impossible roots of a quadratic equation by going out of the line on. which, if real, they would have been laid off. \ B 2 QUATERNIONS. [cHAP. I. His construction is equivalent to the consideration of \/^— 1 as a directed unit-line perpendicular to that on which real quantities are measured. 4. In the usual notation of Analytical Geometry of two dimensions, when rectangular axes are employed^ this amounts to reckoning each unit of length along 0^ as -h a/— T, and on 0^ as --\/— 1 ; while on Ox each unit is -f 1, and on Ox' it is — 1 . K we look at these four lines in circular order, i. e. in the order of positive rotation (opposite to that of the hands of a watch), they give In this series each expression is derived from that which pre- cedes it by multiplication by the factor \/— 1. Hence we may consider \/— 1 as an operator, analogous to a handle perpen- dicular to the plane of xyy whose eflPect on any line is to make it rotate (positively) about the origin through an angle of 90°. 5. In such a system, a point is defined by a single imaginary expression. Thus a'\-h>J—\ may be considered as a single quantity, denoting the point whose coordinates are a and h. Or, it may be used as an expression for the line joining that point with the origin. In the latter sense, the expression a-\-hs/ —X implicitly contains the direction^ as well as the lengthy of this line ; since, as we see at once, the direction is inclined at an angle tan~* - to the axis of a?, and the length is V^a' + i*. Of 6. Operating on this symbol by the factor \/— 1, it becomes — i -h dj \/— 1 ; and now, of course, denotes the point whose x and y coordinates are —J and a; oi* the line joining this point with the origin. The length is still \/fl* -f i*, but the angle the line makes with the axis of x is tan** ( ~ r) ^ ^^^^^ ^ evidently 90° greater than before the operation. SECT. 9.] VECTORS, AND THEIR COMPOSITION. 3 7. De Moivre's Theorem tends to lead us still farther in the same direction. In feet, it is easy to see that if we use, instead of >>/— 1, the more general factor cosa+ \/ — 1 sin a, its effect on any line is to turn it through the (positive) angle a in the plane of of, y, [Of course the former factor, \/— 1, is merely the particular case of this, when a = - •] Thus (cosa-f a/— 1 sina)(«-f i>/— 1) = Acosa— 3sina+ \/ — 1 (a sin a + i cos a), by direct multiplication. The reader will at once see that the new form indicates that a rotation through an angle a has taken place, if he compares it with the common formulae for turning the coordinate axes through a given angle. Or, in a less simple manner, thus — Length = V'(«cosa— Jsina)*-f (asina + Jcosa)' = A/^'-fd* as before. Inclination to axis of or ^ , tan a + - , Asma + ^cosa ^ , a = tan-* r-^ — = tan" «coso— isina , h, 1 tana h ^ = a-f tan-*-- a 8. We see now, as it were, why it happens that (cos a4- a/— 1 sin a)** = cos ma-f >/— 1 sin ma. In fact, the first operator produces m successive rotations in the same direction, each through the angle fi; the second, a single rotation through the angle ma. 9. It may be interesting, at this stage, to anticipate so far as to state that a Quaternion can, in general, be put under the form JV(cosd+tsrsin^), where iV is a numerical quantity, Q a real angle, and tsr»= — 1. B 2 4 QUATBRNIONa [cHAP. I. This expression for a quaternion bears a very close analogy to the forms employed in De Moivre's Theorem ; but there is the essential difference (to which Hamilton's chief invention referred) that cr is not the algebraic \/— 1, but may be any directed unit* line whatever in space. 10. In the present century Argand, Warren, and others, ex- tended the results of Wallis and De Moivre. They attempted to express as a line the product of two lines each represented by a symbol such as a + iv^— 1. To. a certain extent they suc- ceeded, but simplicity was not gained by their methods, as the terrible array of radicals in Warren's Treatise sufficiently proves. IL A very curious speculation, due to Servois, and published in 1813 in Gergonne's Annates y is the only one, so far as has been discovered, in which the slightest trace of an anticipation of Quaternions is contained. Endeavouring to extend to space the form «-f i^/— 1 for the plane, he is glided by analogy to write for a directed unit-line in space the form p cos a-l-5' cos ^ -f r cos y, where a, j3, y are its inclinations to the three axes. He perceives easily that p, q, r must be non-reals : but, he asks, ^^ seraient- elles imaginaires reductibles k la forme generale ^-f jBa/— 1 ?'' This he could not answer. In feet they are the i,j, h of the Quaternion Calculus. (See Chap. II.) 12. Beyond this, few attempts were made, or at least re- corded, in earlier times, to extend the principle to space of thre^ dimensions ; and, though many such have been made within the last forty years, none, with the single exception of Hamilton's, have resulted in simple, practical methods; all, however in- genious, seeming to lead at once to processes and results of fearful complexity. For a lucid, complete, and most impartial statement of the SECT. 14.] VECTORS, AND THEIR COMPOSITION. 5 claims of his predecessors in this field we refer to the Preface to Hamilton's Lectures on Quaternions. 13. It was reserved for Hamilton to discover the use of >/— 1 as a geometric realUy, tied down to no particular direction in space, and this use was the foundation of the singularly elegant, yet enormously powerful. Calculus of Quaternions. "While all other schemes for using \/— 1 to indicate direc- tion make one direction in space expressible by real numbers, the remainder being imaginaries of some kind, leading in general to equations which are heterogeneous ; Hamilton makes all di- rections in space equally imaginary, or rather equally real, thereby ensuring to his Calculus the power of dealing with epace indiflferently in all directions. In fact, as we shall see, the Quaternion method is independent of axes or any supposed directions in space, and takes its refer- «nce lines solely from the problem it is applied to. 14. But, for the purpose of elementary exposition, it is best to begin by assimilating it as closely as we can to the ordinary Cartesian methods of Geometry of Three Dimensions, which are in fact a mere particular case of Quaternions in which most of the distinctive features are lost. We shall find in a little that it is capable of soaring above these entirely, after having employed them in its establishment; and, indeed, as the in- ventor's works amply prove, it can be established, ab initiOy in various ways, without even an allusion to Cartesian Geometry. As this work is written for students acquainted with at least the elements of the Cartesian method, we keep to the first- mentioned course of exposition; especially as we thereby avoid some reasoning which, though rigorous and beautiftil, might be apt, fix)m its subtlety, to prove repulsive to the beginner. We commence, therefore, with some very elementary geo- metrical ideas. 6 QUATERNIONS. [cHAP. L 15. Suppose we have two points A and B in, ipace^ and sup- pose A given^ on how many numbers does W^ relative position depend ? If we refer to Cartesian coordinates (rectangular or not) we find that the data required are the excesses of -B's three coor- dinates over those of A, Hence three numbers are required. Or we may take polar coordinates. To define the moon's position with respect to the earth we must have its Geocentric Latitude and Longitude^ or its Bight Ascension and Declination, and^ in addition^ its distance or radius- vector. Three again. 16. Here it is to be carefiiUy noticed that nothing has been said of the actual coordinates of either A or B^ or of the earth and moon^ in space ; it is only the relative coordinates that are contemplated. Hence any expression^ as AJBy denoting a line considered with reference to direction as well as length, contains implicitly three numbers, and all lines parallel and equal to AB depend in the same way upon the same three. Hence, all lines which are equal and parallel may be represented hy a common symioly and that symbol contains three distinct numbers. In this sense a Une is called a vbctoe, since by it we pass from the one extremity, A, to the other, B] and it may thus be considered as an in- strument which carries A to Bi so that a vector may be em- ployed to indicate a definite translation in space. 17. We may here remark, once for all, that in establishing a new Calculus, we are at liberty to give any definitions what- ever of our symbols, provided that no two of these interfere with, or contradict, each other, and in doing so in Quaternions simplicity and (so to speak) naturalness were the inventor's aim. 18. Let AB be represented by a, we know that a depends on three separate numbers. Now if CD be equal in length to AB SECT; 20.] VECTORS, AND THEIR COMPOSITION. 7 and if these lines be parallel^ we have evidently CD = AB = a, where it will be seen that the sign of equality between vectors contains implicitly equality in length and parallelism in direction. So far we have extended the meaning of an algebraic symbol. And it is to be noticed that an equation between vectors, as o = /3, contains three distinct equations between mere numbers. 19. We must now define -f (and the meaning of — will follow) in the new Calculus. Let Ay By C be any three points and (with the above meaning of =) let lB = ay BC=:py IC=zy. If we define + (in accordance with the idea (§ 16) that a vector represents a translation) by the equation or iB^BC^ACy we contradict nothing that precedes, but we at once introduce the idea that vectors are to be compoundedy in direction and magnitudey like simultaneous velocities, A reason for this may be seen in another way if we remember that by adding the differences of the Cartesian coordinates of A and By to those of the coordinates of B and C, we get those of the coordinates of A and C. 20. But we also see that if C and A coincide (and C may be fl^^y point) AC^Qy for no vector is then required to carry A to C. Hence the above relation may be written, in this case, AB^-BA^Oy or, introducing, and by the same act defining, the symbol — , 52= -iB. 8 QUATERNIONS. [cHAP. I. Hence, the symbol — , applied to a vector , simply shows that its direction' is to be reversed. And this is consistent with all that precedes ; for instance, IB + BC= AC, and A£ = IC-BC, or -IC+CB, are evidently but different expressions of the same truth. 2L In any triangle, ABC, we have, of course, AB + BC+CA = 0; and, in any closed polygon, whether plane or gauche, AB+BC+ -^YZ^ZA^O. In the case of the polygon we have also ZB-f J^-f J^fZ^M These are the well-known propositions regarding composition of velocities, which, by the second law of motion, give u» the geometrical laws of composition of forces. 22. If we compound any number of parallel vectors, the result is obviously a numerical multiple of any one of tbem^ Thus, if A, B, C are in one straight line, BC-xAB; where <r is a number, positive when B lies between A and C, otherwise negative : but such that its numerical value, inde- pendent of sign, is the ratio of the length of BC to that of AB. This is at once evident if AB and BC be commensurable ; and is easily extended to incommensurables by the usual reductio ad ahsurdum. 23. An important, but almost ob^dous, proposition is that any vector may he resolved into three components parallel respectively SECT. 25.] VECTORS, AND THEIR COMPOSITION. 9 to any three given vectors ^ no two of which are parallel, and which are not parallel to one plane. Let OA, OB, OC be the three fixed vectors, OP any other vector. From P draw PQ parallel to CO, meeting the plane BOA in Q. [There mast be a real point Q, else PQ, and therefore CO, would be parallel to BOA, a case specially excepted.] From Q draw QR parallel to BO, meeting OA in R, Then we have 6P=6R + RQ-^QP (§ 21), and these components are respectively parallel to the three given vectors. By § 22 we may express OR as a numerical multiple of OA, RQ o{ OB, and QP of OC, Hence we have, generally, for any vector in terms of three fixed non-coplanar vectors, a, ^, y, "which exhibits, in one form, the three numbers on which a vector depends (§ 16). Here x,y, z are perfectly definite. 24. Similarly any vector, as OQ, in the same plane with OA and OB, can be resolved into components OR, RQ, parallel respectively to OA and OB; so long, at least, as these two vectors are not parallel to each other. 25. There is particular advantage, in certain cases, in employ •. ing a series of three mutually perpendicular unit-vectors as lines of reference. This system Hamilton denotes by i,j, h. Any other vector is then expressible as p = xi'\-yj-^zk. Since iyj, k are unit- vectors, x, y, z are here the lengths of con- terminous edges of a rectangular parallelepiped of which p is the vector-diagonal ; so that the length of p is, in this case, c 10 QUATERNIONS. [CHAP. I. Let or = fi-f ij^ + f* be any other vector, then the vector equation p ^ V obviously involves the following three equations among numbers^ ^ = f , y = n, ^ = f • Suppose i to be drawn eastwards, J northwards, and i upwards, this is equivalent merely to saying that if two points coincide, they are equally to the east {or west) of any third pointy equally to the north {or south) of it, and equally elevated above {or depressed below) its level. 26. It is to be carefiilly noticed that it is only when o, p, y are not eoplanar that a vector equation such as p = Iff, or iPa-fy/3+;2ry = fa-f r/^-f fy, necessitates the three numerical equations For, if o, )8, y be eoplanar (§ 24), a condition of the following form must hold 1 a y = aa-\-op. Hence p = (x'\'Za)a-\-{]/-\-zb)fi, ^= (f + i'«)a + ('7 + f«)/3, and the equation p ■= vr now requires only the two numerical conditions x-\-za:=: (+Ca, y-\-zb = 17+f*. 27. The Commutative and Associative Laws hold in the com-- bination of vectors by the signs + and — . It is obvious that, if we prove this for the sig^ -f , it will be equally proved for — , because — before a vector ($ 20) merely indicates that it is to be reversed before being considered positive. Let A, By Cy B be, in order, the comers of a parallelogram ; we have, obviously. SECT. 29.] VECTORS, AND THEIR COMPOSITION. 11 And AB^SC^ IC= AD-\-BC ^ BC^lB. TIence the commutatiye law is true for the addition of any two vectors, and is therefore generally true. Again, whatever four points are represented by A, B, C, B, we have AB = AB-^BB = IC+ CB, or ZB-f JC'+CS = AB-\-{BC+CB) = {AB^BC)+CB. And thus the truth of the associative law is evident. 28. The equation ^ /> = ^P> where p is the vector connecting a variable point with the origin, P a definite vector, and x an indefinite number, represents the straight line drawn from the origin parallel to /3 (§ 22). The straight line drawn from J, where OA = a, and parallel to j3, has the equation p = a+^j3 (1) In words, we may pass directly from to P by the vector OP OT p; or we may pass first to A, by means of OA or o, and then to P along a vector parallel to )3 (§ 16). Equation (1) is one of the many usefiil forms into which Quaternions enable us to throw the general equation of a straight line in space. As we have seen (§ 25) it is equivalent to tAree numerical equations ; but, as these involve the indefinite quan- tity a?, they are virtually equivalent to but two, as in ordinary Geometry of Three Dimensions. 29. A good illustration of this remark is furnished by the fact that the equation p =ya-f-a?/3, which contains two indefinite quantities, is virtually equivalent to only one numerical equation. And it is easy to see that it represents the plane in which the lines a and fi lie ; or the C 2 • 12 QUATERNIONS. [chap. I. surface which is formed by drawing, through every point of OA, a line parallel to OJB. In fact, the equation, as written, is simply § 24 in symbols. And it is evident that the equation is the equation of a plane passing through the extremity of y, and parallel to a and j3. It will now be obvious to the reader that the equation P =i?iai+J»aa2+ = 2i?a, where Oi, aa, &c. are given vectors, and jOi, j»2, &c. numerical quantities, represe^its a straight line if jOj, p^y &c. be linear functions of one indeterminate number ; and a plane, if they be linear expressions containing two indeterminate numbers. Later (§31 (/)), this theorem will be much extended. 30. The equation of the line joining any two points A and B^ where 0A= a and OB = j8, is obviously p = a + ^(i3 — a), or /) =^-fy(a-/3). These equations are of course identical, as may be seen by putting 1 — y for x. The first may be written p + {X'-l)a—xp = 0; or i?p + ^a + r/3 = 0, subject to the condition p-{-q-\-r = identically. That is — A homogeneous linear function of three vectors, equated to zero, expresses that the extremities of these vectors are in one straight line, if the sum of the coefficients be identically zero. Similarly, the equation of the plane containing the extremities A) By C of the three non-coplanar vectors a, j8, y is p = a+ir(/3-a)4-j^(y-^), where x and y are each indeterminate. SECT. 31.] VECTORS, AND THEIR COMPOSITION. 13 This may be written ^ with the identical relation p-^q + r-hs=0, which is the condition that four points may lie in one plane. 3L We have already the means of proving, in a very simple manner, numerous classes of propositions in plane and solid geometry. A very few examples, however, must suffice at this stage; since we have hardly, as yet, crossed the threshold of the subject, and are dealing with mere linear equations con- necting two or more vectors, and even with them we are re- stricted as yet to operations of mere addition, (a^ The diagonals of a parallelogram bisect each other. Let ABCB be the parallelogram, the point of intersection of its diagonals. Then which gives AO-OC^iob-OB, The two vectors here equated are parallel to the diagonals re- spectively. Such an equation is, of course, absurd unless (1) The diagonals are parallel, in which case the figure is not a parallelogram ; (2) Id = 6C, and BO =: OB, the proposition. (i.) To show that a triangle can he constructed, whose sides are parallel, and equal, to the bisectors of the sides of any triangle. Let ABC be the triangle, Aa, Bb, Cc the bisectors of the sides. Then : A 14 QUATERNIONS. [OMAr. I. 5« . . - =Sc+ic2, tfc . - - =CA-^iA£. Hence ia+5i + ^= |(i5+^+C5) = 0; ;! * which (§21) proves the proposition. ] Also 2^ = ZB + i^ ' _ _ _ ^ = i(i5-(7T) = iilS-i-IC), results which are sometimes useful. They may be easily veri^ by producing Aa to twice its length and joining the extrei^ with £, t ' (i'.) Tke bisectors of the sides of a triangle meet in a poi$i^ which trisects each of them. Taking A as origin, and putting a, )8, y for vectors parallelj^ and equal, to the sides taken in order; the equation of Bb is (§ 28 (1)) p = y + J?(y + -|.) = (H.a;)y + -|-^. That of Cc isj in the same way, P=-(l+^)^-fy. At the point 0, where Bb and Cc intersect, p = (l+a;)y + |^=-(H-y)^-|y. Since y and /3 are not parallel, this equation gives • l+ar = -|, and | = -(1+^). From these ar = y = — § . Hence iO = i (y-^S) = %Ia. (See Ex. {b).) SECT. 31.] VECTORS, AND THEIR COMPOSITION. 15 This equation shows, being a vector one, that Aa passes through Oy and that A0\ Oa\\2\\. (c.) If 62= a, dB=z p, 6C= aa-^-b^, be three given co-planar vectors, and the lines indicated in the figure be drawn , the points a, )3, y lie in a straight line. We see, at once, by the process indicated in § 30, that Oc= ^ ^ Hence we easily find 6P Oa = - TU- Ofi = aa 1 — 2a— i' Oy = — ga + 3/3 b—a 1— (»- These give — (l-.dj-.2«)0^ + (l-2«-.*)0^-(5-fl)Oy = 0. But —(l*-a—2*)-h(l—2«— *)—(*— a) = identically. This, by § 30, proves the proposition. (d.) Let OA = a, OJB = j3, be any two vectors. If MP be parallel to OB ;- and OQ, BQ, be drawn parallel to APy OP respectively ; the locus of Q is a straight line parallel to OA. Let OM = ea. Then Hence the equation of OQ is 9 =^(^^a-hiP/3); and that of BQ is p = &^z{ea'\-xfi). 16 QUATERNIONS. [cHAP. I. At Q we have, therefore, Xy=: \-\-ZXy ^ y{€-\)^ze,) These give xy -z^ e, and the equation of the locus of Q is P = ^^+/a, i. e. a straight line parallel to OA, drawn through N in OB produced, so that ON '.OB 11 OMiOA. CoE. If BQ meet MP inq,Pq = fi; and if JP meet NQ mp, Also, for the point R we have pR = AF, QR = Bq. Hence if from any two points, A and By lines be drawn intercept^ ing a given length Pq on a given line Mq ; and if, from R their point of intersection, Rp be laid off = PA, and RQ ^ qB ; Q and p lie on a fixed straight line, and the length of Qp is constant. (e.) To find the centre of inertia of any system. If OA = a, OB = ai, be the vector sides of any triangle, the vector from the vertex dividing the base AB in C so that BC : CA :: mim^ is ma-{-Miai For AB is ai — a, and therefore ^Cis Ml {a^-a). Hence 00^61+ AC SECT. 31.] VECTORS, AND THEIR COMPOSITION. 17 This expression shows how to find the centre of inertia of two masses ; m at the extremity of a, m,^ at that of aj. Introduce m^ at the extremity of a^, then the vector of the centre of inertia of the three is, by a second application of the formula. m + nii -)+«^2as ma+niiai-hm^ai For any number of masses, expressed generally by m at the extremity of the vector a, we have the vector of the centre of inertia ' _/ \ _ 2 (ma) This may be written 2m(a--/3) = 0. Now ai--)3 is the vector of m^ with respect to the centre of inertia. Hence the theorem, If the vector of each element of a mass, drawn from the centre of inertia, be increased in length in proportion to the mass of the element, the sum of all these vectors is zero, {f) We see at once that the equation P^at^^, where ^ is an indeter- minate number, and a, j3 given vectors, represents a parabola. The origin, 0, is a point on the curve, /3 is parallel to the axis, i. e. is the diameter OB drawn from the origin, and a is OA the tangent at the origin. In the figure 18 QUATERNIONS. [CHAP. I. The secant joining the points where t has the values t and l^ is represented by the equation p = at+^-^x{at'-^^-at-^) (§30) = at + ^ + x{€-(){a+p^^}. Put f::^tf and write x {or x{f'~() [which may have any value] and the equation of the tangent at the point {f) is rva x=—t, p = — , or the intercept of the tangent on the diameter is — the abscissa of the point of contact. Otherwise: the tangent is parallel to the vector a+^t or at-^pt^ or a^ + ^ + ^or OQ+OP. But?y=fO+QP, hence TO ^ OQ. {0.) Since the equation of any tangent to the parabola is let us find the tangents which can be drawn from a given point. Let the vector of the point be p=^a+?/3 (§24). Since the tangent is to pass through this point, we have, as conditions to determine t and x, t+x =jp, — -\^xt±zq; by equating respectively the coefficients of a and p. Hence ^ =jo + >/jo* — 25'. SECT. 31.] VECTORS, AND THEIR COMPOSITION. 19 Thus, in general, two tangents can be drawn from a given point. These coincide if P' = 2q; that is, if the vector of the point from which they are to be drawn is p^ i. e. if the point lies on the parabola. They are imaginary if 2 q >jt?'», i. e. if the point be r heing positive. Such a point is evidently within the curve, as at R, where OQ = ^/3, QP ^pa, PB = rp. Ih,) Calling the values of t for the two tangents found in (^) t^ and t^ respectively, it is obvious that the vector joining the points of contact is which is parallel to t +t or, by the values of t^ and t^ in (^), o+joj8. Its direction, therefore, does not depend on q. In words, If pairs of tangents be drawn to a parabola from points of a diameter produced y the chords of contact are parallel to the tangent at the vertex of the diameter. This is also proved by a former result, for we must have OT ior each tangent equal to QO. (i.) The equation of the chord of contact, for the point whose vector is p=pa + q^y Bt ^ is thus p = a^i+-y--fy(a-fj»^).. D 2 20 QUATERNIONS. . [chap. I. Suppose this to pass always through the point whose vector is p = aa-\-b^. Then we must have ^1 4-y = a, or ^1 =jo f >yj»* — 2j9fl + 2i. Comparing this with the expression in (^), we have that is^ the point from which the tangents are drawn has the vector = —bfi^p{a-\-a^), a straight line (§ 28 (1)). The mere form of this expression contains the proof of the usual properties of the pole and polar in the parabola; but, for the sake of the beginner, we adopt a simpler, though equally general, process. Suppose a = 0, This merely restricts the pole to the par- ticular diameter to which we have referred the parabola. Then the pole is Q, where p =r bB- and the polar is the line TU, for which p =-'bP'\-pa. Hence tke polar of any point is parallel to the tangent at the extremity of the diameter on which the point lies, and its inter^ section with that diameter is as far beyond the vertex as the^poU is within, and vice versa. (J,) As another example let us prove the following theorem. I/^ a triangle be inscribed in a parabola, the three points in which the sides are met by tangents at the angles He in a straight line. Since is any point of the curve, we may take it as jone SECT. 31.] . VECTOES, AND THEIR COMPOSITION. 21 corner of the triangle. Let t and t^ determine the others. Then, if w^, Wa, m^ represent the vectors of the points of inter- section of the tangents with the sides, we easily find O-a p . 2t-t .(. + %^) These vahies give «^3 a. U, -t 2t-t^ t^- '^P U ' t ' tt. CTa = 0. Also -— — — ^ — — = identically. Hence, by § 30, the proposition is proved. (^.) Other interesting examples of this method of treating curves will, of course, suggest themselves to the student. Thus p = a cos ^ -f j3 sin ^ or p = ax-\-fi\/\^x'^ represents an ellipse, of which the given vectors a and /3 are semi-conjugate diameters. Again, pc=at-^y or p = atana? + /3coti?? t evidently represents a hyperbola referred to its asymptotes. But, so far as we have yet gone, as we are not prepared to determine the lengths or inclinations of vectors, we can only investigate a very small class of the properties of curves, re- presented by such equations as those above written. (L) We may now, in extension of the statement in § 29, make the obvious remark that p = 2pa 22 QUATERNIONS. [CHAP. I. is the equation of a curve in space^ if the numbeife j»i, jOj, &e. are functions of one indeterminate. In such a ease the equation, is sometimes written ^ _ a^u)^ But, i(jpi,pty &e. be functions of two indeterminates, the locus of the extremity of p is a surface ; whose equation is sometimes written ^^^(j^^^y (m.) Thus the equation p = acos^ + /3sin^-|-y^ belongs to a helix. Again, p =J»a + y^ + ry with a condition of the form ajo^-f ^j»-f (?r^ = 1 belongs to a central surface of the second order, of which a, j8, y are the directions of conjugate diameters. If «, i, c be all positive, the surface is an ellipsoid. 32. In Example {/) above we performed an operation equi- valent to the diflPerentiation of a vector with reference to a single numerical variable oT which it was given as an explicit fimction. As this process is of very great use, especially in quaternion investigations connected with the motion of a particle or point ; and as it will aflPord us an opportunity of making a preliminary step towards overcoming the novel difficulties which arise in quaternion differentiation; we will devote a few sections to a more careful exposition of it. 33. It is a striking circumstance, when we consider the way in which Newton^s original methods in the Differential Calculus have been decried, to find that Hamilton was obliged to employ them, and not the more modem forms, in order to overcome the characteristic difficulties of quaternion diflferentiation. Such a thing as a differential coefficient has absolutely no meaning in SECT. 35.] VECTORS, AND THEIR COMPOSITION. 23 quaternions, except in those special cases in which we are dealing with degraded quaternions, such as numbers, Cartesian coor- dinates, &c. But a quaternion expression has always a differ^ entialy which is, simply, what Newton called 2i fluxion. As with the Laws of Motion, the basis of Dynamics, so with the foundations of the Differential Calculus ; we are gradually coming to the conclusion that Newton's system is the best after all. 34. Suppose p to be the vector of a curve in space. Then, generally, p . may be expressed as the sum of a number of terms, each of which is a multiple of a given vector by a function of some one indeterminate ; or, as in § 3 1 (/), if P be a point on the curve, 6P = pz=<p{f). And, similarly, if Q be any other point on the curve, OQ = p, = <p{t,)^<l>{t^btl where bt is any number whatever. The vector-chord PQ is therefore, rigorously, hp = pi — p = <^(t-^ht) — ^t, _ 35. It is obvious that, in the present case, because the vectors involved in (j> are constant, and their numerical multipliers alone vary, the expression <f){t-{-hf) is, by Taylor's Theorem, equivalent to ^(^ + -__8^+_^__ + Hence; ,^^mu+'-l^ ^ ^ ^o. ^ dt dp 1.2 And we are thus entitled to write, when ht has been made in- definitely small. In such a case as this, then, we are permitted to diflferentiate. 24 QUATERNIONS. [chap. I. or to fonn the diflferential coefficient of, a vector according to the ordinary rules of the Differential Calculus. But great ad- ditional insight into the process is gained by applying Newton^s method. 36. Let OP be and OQi where dt is any number what- ever. The number t may here be taken as representing timey i. e. we may suppose a point to move along the curve in such a way that the value of t for the vector of point P of the curve de- notes the interval which has elapsed (since a fixed epoch) when the moving point has reached the extremity of that vector. If, then, dt represent any interval, finite or not, we see that 6q,^it>{t-\-dt) will be the vector of the point after the additional interval dt. But this, in general, gives us little or no information as to the velocity of the point at J^. We shall get a better approxi- mation by halving the interval dty and finding Q^, where OQa = <^(^4-4rf^, as the position of the moving point at that time. Here the vector virtually described in i^/J is PQa. To find, on this supposition, the vector described in dt, we must double this, and we find, as a second approximation to the vector which the moving point would have described in time dt, if it had moved for that period in the direction and with the velocity it had at P, P^, = 2PQ, = 2(OQ,-aP) = 2{<^(^ + i^0-<^^}- SBCT. 38.] VECTORS, AND THEIR COMPOSITION. 25 The next approximation gives ^3 = 3PQ, = 3(OQ3-QP) And so on^ each step evidently leading ns nearer the sought truth. Hence, to find the vector which would have been described in time dt had the circumstances of the motion at P remained undisturbed, we must find the value of We have seen that in this particular case we may use Taylor's Theorem. We have, therefore, dp = ^x=«^{*'W^rf^+*''(0^ ^ +&e.} = 4>\()dt. And, if we choose, we may now write 37. But it is to be most particularly remarked that in the whole of this investigation no regard whatever has been paid to the magnitude of dt. The question which we have now answered may be put in the form — A point describes a given curve in a given manner. At any paint of its path its motion suddenly ceases to be accelerated. What space will it describe in a definite interval ? As Hamilton well observes, this is, for a planet or comet, the case of a '' celestial Attwood's machine.^' 38. If we suppose the variable, in terms of which p is ex- pressed, to be the arc, «, of the curve measured from some fixed point, we find as before Hi = ^'(s)ds. E 26 QUATERNIONS. [CHAP. I- From the veiy nature of the question it is obvious that the length of dp must in this case be ds. This remark is of im- portance^ as we shall see later ; and it may therefore be useful to obtain afresh the above result without any reference to time or velocity. 89. Following strictly the process of Newton's Vllth Lemma, let us describe on Pq^ an arc similar to PQ^, and so on. Then obviously, as the subdivision of ds is carried farther, the new arc (whose length is always ds) more and more nearly coincides with the line which expresses the corresponding approximation to dp, 40. As a final example let us take the hyperbola p = at+^. Here . fi. This shews that the tangent is parallel to the vector In words, if ike vector {from the centre) of u point in a hyperbola be one diagonal of a parallelogram^ two of whose sides coincide with the asymptotes^ the other diagonal is parallel to the tangent at the point. 41. Let us reverse this question, and seek the envelop of a Une which cuts off from two fixed axes a triangle <f constant area. If the axes be in the directions of a and /8, the intercepts may evidently be written at and y . Hence the equation of the line is (§ 30) p = at-^xy^ — aty <t The condition of envelopment is, obviously, dp = 0. SECT. 42.] VECTORS, AND THEIR COMPOSITION. 27 This gives = {a-a?(-^ + a)}dt-^(^-at)dx. [We are not here to equate to zero the eoeffieienta of dt and dx ; for we must remember that this equation is of the form =i?a + g'/3, wliere j9 and q are numbers ; and that^ so long as a and j3 are actual and non-parallel vectors^ the existence of suoh an equation requires i? = 0, ? = 0.] Hence {l'-x)dt'-tdx = 0, and —^^^j. _ = 0. t* ^ t From these, at once, x-^^, since dx and dt are indeterminate. Thus the equation of the envelop is the hyperbola as before ; o, jS being portions of its asymptotes. 42. It may assist the student to a thorough comprehension of the above process, if we put it in a slightly different form. Thus the equation of the enveloping line may be written p = aj{(l-a?)+i3|, which gives dp = = ad{t{\ —a?)) +/3rf (-) . Hence, as a is not parallel to j3, we must have and these are, when expanded, the equations we obtained in the preceding section. E % 28 QUATBRNIOKS. [ohap. L 43. For farther illastration we give a solution not directly employing the differential calcnius. The equations of any two of the enveloping lines are 3 t and t^ being given^ while s and y are indeterminate. At the point of intersection of these lines we have (§ 26), ^(l-ar) = ^,(l-y),- ^ — JL t t^ * These give, by eliminating y, t or X "^ • Hence the vector of the point of intersection is and thus, for the ultimate intersections, where j^ —- = 1, 1 p = \{at-\-^ as before. CoE. (1). If «, = 1, /, = — , or the intersection lies in the diagonal of the parallelogram on a,/3. Cor. (2). If t^ = mtj where m is constant, SECT. 44.] VECTORS, AND THEIR COMPOSITION. 29 But we have also x = — • Hence the locus of anoint which divides in a given ratio a line cutting off a given area from two fixed axes, is a hyperbola of which these axes are the asymptotes. Cor. (3). If we take tt^\t'\-t^ = constant the locus is a parabola ; and so on. 44. The reader who is fond of Anharmonic Ratios and Trans- versals will find in the early chapters of Hamilton's Elements of (Quaternions an admirable application of the composition of vectors to these subjects. The Theory of Geometrical Nets, in a plane, and in space, is there very folly developed; and the method is shewn to include, as particular cases, the processes of Orassmann's Ausdehnungslehre and Mobius' Baryeentrische CalcuL Some very curious investigations connected with curves and sar&ces of the second and third orders are also there founded apon the composition of vectors. EXAMPLES TO CHAPTER I. 1. The lines which join, towards the same parts, the ex- Iremities of two equal and parallel lines are themselves equal and parallel. {Huclid, I. xxxiii.) 2. Knd the vector of the middle point of the line which joins the middle points of the diagonals of any quadrilateral, plane or gauche, the vectors of the comers being given ; and so prove that this point is the mean point of the quadrilateral. 30 QUATERNIONS. [cHAP. I. If two opposite sides be divided proportionally, and two new quadrilaterals be formed by joining the points of division, the mean points of the three quadrilaterals lie in a straight line. Shew that the mean point may also be found by bisecting the line joining the middle points of a pair of opposite sides. 3. Verify that the property of the coefficients of three vectors whose extremities are in a line* (§ 30) is not interfered with by altering the origin. 4. If two triangles ABC, abc, be so situated in space that Jjdf Bb, Cc meet in a point, the intersections of AB^ ab, of BC, be, and of CA, ca, lie in a straight line. 5. Prove the converse of 4, i. e. if lines be drawn, one in each of two planes, from any three points in the straight line in which these planes meet, the two triangles thus formed are sections of a common pyramid. 6. If five quadrilaterals be formed by omitting in succession each of the sides of any pentagon, the lines bisecting the diagonals of these quadrilaterals meet in a point. (H. Fox Talbot.) 7. Assuming, as in § 7, that the operator cos^+ V—l sinO turns any radius of a given circle through an angle in the positive direction of rotation, without altering its length, deduce the ordinary formulae for cos {A-{-B), cos (A^B), sin {A-\-B), and sin {A—B)y in terms of sines and cosines of A and B. 8. If two tangents be drawn to a hyperbola, the line joinmg the centre with their point of intersection bisects the lines join- ing the points where the tangents meet the asymptotes : and the tangent at the point where it meets the curves bisects the intercepts of the asymptotes. 9. Any two tangents, limited by the asymptotes, divide each other proportionally. CHAP. L] EXAMPLES TO CHAPTER I. 31 10. If a chord of a hyperbola be one diagonal of a paralM- ogram whose sides are parallel to the asymptotes, the other diagonal passes through the centre. 11. Show that is the equation of a cone of the second degree, and that its section by the plane is an ellipse which touches, at their middle points, the sides of the triangle of whose corners a, j3, y are the vectors. (Hamilton, Elements y p. 96.) 12. The lines which divide, proportionally, the pairs of opposite sides of a gauche quadrilateral, are the generating lines of a hyperbolic paraboloid. (Ibid, p. 97.) 13. Show that where x-^y-\-z = 0, represents a cone of the third order, and that its section by the plane is a cubjc curve, of which the lines are the asymptotes and the three (real) tangents of inflexion. Also that the mean point of the triangle formed by these lines is a conjugate point of the curve. Hence that the vector ^-Hi3 + y is a conjugate ray of the cone. {Ibid. p. 96.) CHAPTER 11. PRODUCTS AND QUOTIENTS OF VECTORS. 46. Xl/E now come to the consideration of points in ▼^ which the Calculus of Quaternions differs entirely from any previous mathematical method; and here we shall get an idea of what a Quaternion is^ and whence it derives its name. These points are Aindamentally involved in the novel use of the symbols of multiplication and division. And the simplest in- troduction to the subject seems to be the consideration of the quotient^ or ratio^ of two vectors. 46. If the given vectors be parallel to each other^ we have already seen (§ 22) that either may be expressed as a numerical multiple of the other; the multiplier being simply the ratio of their leng^hs^ taken positively if they are similarly directed^ negatively if they run opposite ways. 47. If they be not parallel^ let OA and OB be drawn parallel and equal to them from any point 0; and the question is re- duced to finding the value of the ratio of two vectors drawn from the same point. Let us try to find tipon how many^ distinct numbers this ratio depends* We may suppose OA to be changed into OB by the following processes. 1st. Increase or diminish the length of OA till it becomes equal to that of OB, For this only one number is required, viz. the ratio of the lengths of the two vectors. As Hamilton remarks, this is a positive, or rather a signless^ number. 2nd. Turn OA about until its direction coincides with that of OB, and (remembering the effect of the first SECT. 49.] PRODUCTS AND QUOTIENTS OP VECTORS. 33 operation) we see that the two vectors now coincide or become identical. To specify this operation three more numbers are required^ viz. two angles (such as node and inclination in the case of a planet's orbit) to fix the plane in which the rotation takes place^ and one angle for the amount of this rotation. Thus it appears that the ratio of two vectors, or the multiplier required to change one vector into another, in general depends w^oufoTJi/r distinct numbers, whence the name quaternion. 48. It is obvious that the operations just described may be performed, with the same result, in the opposite order, being perfectly independent of each other. Thus it appears that a quaternion, considered as the factor or agent which changes one definite vector into another, may itself be decomposed into two factors of which the order is immaterial. The stretching factor, or that which performs the first opera- tion in § 47, is called the Tensor, and is denoted by prefixing T to the quaternion considered. The turning &ctor, or that corresponding to the second opera^ tion in § 47, is called the Vbbsor, and is denoted by the letter U prefixed to the quaternion. 49. Thus, if 0A = a, OB = /8, and if j be the quaternion which changes o to )8, we have which we may write in the form ^ = ?> or pa" = q, if we agree to defifie that -.a = i8a-».a = /8. a Here it is to be particularly noticed that we write q before a to signify that a is multiplied by q, not q multiplied by o. [This remark is of extreme importance in quaternions, for. 34 QUATERNIONS. [cHAP. II. as we shall soon see^ the Commutative Law does not generally apply to the factors of a product.] We have also, by § 47, q^TqUq^UqTq, where, as before, Tq depends merely on the relative lengths of a and j3, and Uq depends solely on their directions. Thus, if tti and p^ be vectors of unit length parallel to a and ^ respectively, R R a. at a 50. We must now carefully notice that the quaternion which is the quotient when fi is divided by a in no way depends upon the absolute lengths, or directions, of these vectors. Its value will remain unchanged if we substitute for them any other pair of vectors which (1) have their lengths in the same ratio, (2) have their common plane the same or parallel, and (3) make the same angle with each other. Thus in the annexed figure if, and only if, ,^. 0,B, OB (2) plane A OB parallel to plane A,0,B,, (3) lAOB- LA^O^By, [Equality of angles is understood to in- clude similarity in direction. Thus the ro- tation about an upward axis is negative (or right-handed) from OA to OBy and also from O^A^ to OyB^:\ SECT. 63.] PRODUCTS AND QUOTIENTS OF VECTORS. 35 5L The Reciprocal of a quaternion q is defined by the equation, Hence if - = ^, or we must have - = - = g-^. For this gives ^ • ^ = S^^ • ?^> P and each side is evidently equal to a. Or, we may reason thus, q changes OA to OB, qr^ must there- fore change 0£ to OA, and is therefore expressed by - (§ 49). The tensor of the reciprocal of a quaternion is therefore the reciprocal of the tensor ; and the versor differs merely by the reversal of its representative angle. 52. The Conjugate of a quaternion q, written Kq, has the same tensor, plane, and angle, only the angle is taken the reverse way. Thus, if OA'^OA, and lAOB^ LAOB, \B OB OB . \ r JT -T= = a, ^r^ir- = coniuffate oi q^Kq, OA ^ OA' ^ ^ By last section we see that Kq.= {Tqyq-\ Hence qKq = Kq,q = {Tq)K This proposition is obvious, if we recollect that the tensors of q and Kq are equal, and that the versors are such that either reverses the effect of the other. The joint effect of these factors is therefore merely to multiply twice over by the common tensor. 63. It is evident from the results of § 50 that, if a and /3 F % 36 QUATERNIONS. [chap. II. be of equal length, their quaternion quotient becomes a versor (the tensor being unity) and may be represented indifferently by any one of an infinite number of arcs of given length lying on the circumference of a circle, of which the two vectors are radii. This is of considerable importance in the proofs which follow. Thus the versor -r= may be repre- sented by the arc AB, which may be written AB, OB And, similarly, the versor ^ is represented by A^B^ which is equal to (and measured in the same direction as) AB if LAfiB^^ LAOB, i. e. if the versors are equal. 54. By the aid of this process, when a versor is represented by an arc of a great circle on the unit-sphere, we can easily prove that quaternion multiplication is not generally commutative. Thus let q be the versor AB or OB "^ '-^ -^=T, Make BC-=- AB, then q may OA OC also be represented by -r^r In the same way any other versor r may be represented by DB or BE ^^ OB 6E and by -= or -n=r . ^ OB OB The line 0^ in the figure is definite, and is given by the intersection of the planes of the two versors ; being the centre of the unit-sphere. , Now rOD = OB, and qOB = OC, Hence qrOB^OC, SECT. 58.] PRODUCTS AND QUOTIENTS OF VECTORS. 37 OC ^-N or qr = -=■ , and may therefore be represented by the are JDC of a great circle. But rq is easily seen to be represented by the arc AE, For qOA= OB, and r05 = OE, QJLl whence rqOA = OE, and rq = -==■• OA Thus the versors rq and qr, though represented by arcs of equal length, are not generally in the same plane and are therefore unequal : unless the planes of q and r coincide. Calling OA a, we see that we have assimied, or defined, in the above proof, that q.ra = qr.a and r,qa = rq,a when qa, ra, qra, and rqa are all vectors. 55. Obviously C£ is Kq, BB is Kr, and CB is K{qr). But CB^BB.CBy which gives us the very important theorem K{qr) = Kr.Kq. 56. The propositions just proved are, of course, true of quaternions as weU as of versors ,• for the former involve only an additional numerical factor which has reference to the length merely, and not the direction, of a vector (§ 48). 57. Seeing thus that the commutative law does not in general hold in the multiplication of quaternions, let us enquire whether the Associative Law holds. That is, if jo, j, r be three quaternions, have we p .qr •=. ;pq,rl This is, of course, obviously true if jo, q, r be numerical quantities, or even any of the imaginaries of algebra. But it cannot be considered as a truism for symbols which do not in general give 58. In the first place we remark that jo, q, and r may be considered as versors only, and therefore represented by arcs of 38 QUATERNIONS. [chap. II. great circles, for their tensors may obviously /(§ 48) be divided out from both sides, being commutative with the versors. Let AB =jo, m)=^c2 = q,2JidI^=r. Join BC and produce the great circle till it meets JEF in ^, BXidmake KH=F£=r,mdHG = CB=pq (§54). Join GK. Then KUSl.m^s KG^UG.KH=pq.r. Join FD and pro- duce it to meet AS in M, Make LM^FD, and MN=AB, ^^ and join NL, Then LN=MN.LM=p.qr, Hence to shew that p.qr =pq,r all that is requisite is to prove that LN, and KG, described as above, are equal arcs of the same great circle, since, by the figure, the}'^ are evidently measured in the same direction. This is perhaps most easily effected by the help of the fundamental properties of the curves known as Spherical Conies, As they are not usually familiar to students, we make a slight digression for the purpose of proving these fundamental properties; after Chasles, by whom and Magnus they were discovered. An independent proof of the associative principle will presently be indicated, and in Chapter VII we shall employ quaternions to give an independent proof of the theorems now to be established. 59.* Dbf. a spherical conic is the curve of intersection of a cone of the second degree with a sphere, the vertex of the cone being the centre of the sphere. If a cone have one series of circular sections, it has another series, and any two circles belonging to different series lie on a sphere. This is easily proved as follows. Describe a sphere. A, cutting the cone in one circular section. SECT. 59.] PKODUCTS AND QUOTIENTS OF VECTORS. 89 C, and in any other point whatever, and let the side OpP of the cone meet A in p, P ; P being a point in C. Then PO'Op is constant, and, therefore, since P lies in a plane, p lies on a sphere, «, passing through 0. Hence the locus, c, of js is a circle, being the intersection of the two spheres A and a. Let OqQ be ^ny other side of the cone, q and Q being points in c, C, respectively. Then the quadrilateral q QPp is inscribed in a circle (that in which its plane cuts the sphere A) and the exterior angle at p is equal to the interior angle at Q. If OX, OM be the lines in which the plane POQ cuts the cyclic plants (planes through parallel to the two series of circular sections) they are obviously parallel to pq, QP^ respectively ; and therefore lLOp = lOpq=lOQPz= IMOQ. Let any third side, OrE, of the cone be drawn, and let the plane OPS cut the cyclic planes in 01, Om respective- ly. Then, evi- dently, 110L= A qpr, L MOm = Z QPR, and these angles are independent of the position of the points p and Py if Q and R be lixed points. In a section of the above diagram by a sphere whose centre is 0, IL, Mm are the great circles which re- present the cyclic planes, PQR is the spherical conic which represents the cone. The point P represents the line OpP, and so with the others. 40 QUATERNIONS. [cHAP. II. The propositions above may now be stated thus ArcPi = arcJfQ; and, if Q and E be fixed, Mm and IL are constant arcs whatever be the position of P. 60. The application to § 58 is now obvious. In the fig^e of that article we have PE^KH, ED^CA, SO^CB, LM= FD. Hence i, C, G, D are points of a spherical conic whose cycKc planes are those of AB, FE, Hence also KG passes through L, and with LM intercepts on AB an arc equal to AB. That is, it passes through N, or KG and LN are arcs of the same great circle : and they are equal, for G and L are points in the spherical conic. Also, the associative principle holds for any number of quater- nion &ctors. For, obviously, qr,8t=qrs,t, &c., &c.^ since we may consider ^r as a single quaternion, and the above proof applies directly. 6L That quaternion addition, and therefore also subtraction, is commutative, it is easy to shew- For if the planes of two qua- ternions, q and r, intersect in the line OA^ we may take any vector OA in that line, and at once find two others, O^and 067, such that 6B=q6Ay and 00 = rOA. And {q+r)6A = 6B+0C= 0C+ OB = (r+j)Q2, since vector addition is commutative (§ 27). Here it is obvious that {q-^-r) OAy being the diagonal of the parallelogram on OB, OC, divides the angle between OB and OC SBCfT. 62.] PRODUCTS AND QUOTIENTS OP VECTORS. 41 in a ratio depending solely on the ratio of the lengths of these lines^ i. e. on the ratio of the tensors of q and r. This will be usefiil to us in the proof of the distributive law, to which we proceed. 62. Quaternion multiplication, and therefore division, is dis- tributive. One simple proof of this depends on the possibility, shortly to be proved, of representing any quaternion as a linear function of three given rectangular unit-vectors. And when the proposition is thus established, the asso- ciative principle may readily be deduced from it. But we may employ for its proof the proper- ties of Spherical Conies already employed in de- monstrating the truth of the associative principle. For continuity we give an outline of the proof by this process. Let £Ay CA represent the versors of q and r, and be the great circle whose plane is that of J9. Then, if we take as operand the vector OJ, it is obvious that U[qj^f^ will be represented by some such arc as DA where B,D, C sj^e in one great circle; for {q-^-rjOA is in the same plane as qOA and rOA, and the relative magnitudes of the arcs BS and DC depend solely on the tensors of q and r. Produce BA, DA, CA to meet be in i, d, c respectively, and make M = £A, Fd=:DA, Gh = CA. Also make J^=58=cy =j». Then B, P, G, A lie on a spherical conic of which BC and be are the cyclic arcs. And, because ij3=35=cy, j3^, bF, yG, when produced, meet in a point H 42 QUATERNIONS. [cHAP. II. which is also on the spherical conic (§ 59*). Let these arcs meet BC mJyL,K respectively. Then we have Also U^BI), and KL=^CI>. Andj on comparing the portions of the figure bounded respect- ively by HKJ and by ACB we see that (when considered with reference to their effects as factors multiplying OH and OA respectively) P ^(?+ bears the same relation to pUq and p Ur that U{q+r) bears to Uq and Ur. But T{q + r) U{q + r) = q + r = TqUq^-Trllr. Hence T{q -f r),p U{q + r) = Tq.p Uq-^-Tr.p Ur ; or, since the tensors are mere numbers and commutative with all other factors, p{q-\'r) —pq+pr. In a similar manner it may be proved that {q + r)p ^zqp-^-rp. And then it follows at once that {p-\-q){r-\-8) ^pr-^-ps+qr+qs. 63. By similar processes to those of § 53 we see that versors, and therefore also quaternions, are subject to the index- law ^^ ^„ _ ^^+^^ at least so long as m and n are positive integers. The extension of this property to negative and fractional «ECT. 66.] PEODUCTS AND QUOTIBliirTS OF VECTORS. 43 exponents must be deferred until we have defined a negative ^r fractional power of a quaternion. 64. We now proceed to the special case of quadrantal versors, from whose properties it is easy to deduce all the foregoing results of this chapter. These properties were indeed those whose discovery by Hamilton in 1843 led almost intuitively to the establishment of the Quaternion Calculus. We shall eontemb ourselves at present with an assumption, which will be shewn to lead to consistent results ; but at the end of the chapter we shall shew that no other assumption is possible, following for this purpose a very curious quasi-metaphysical speculation of Hamilton's. 65. Suppose we have a system of three mutually perpen- dicular unit-vectors, drawn from one point, which we may call for shortness 7, /, K, Suppose also that these are so situated that a positive (i. e. left-handed) rotation through a right angle about I as an axis brings / to coincide with K, Then it is obvious that positive quadrantal rotation about / will make K coincide with I; and, about K, will make I coincide with /. For definiteness we may suppose / to be drawn eastwards^ J northwards y and K upwards. Then it is obvious that a positive (left-handed) rotation about the eastward line (7) brings the northward line (/) into a vertically upward position {K) ; and so of the others. 66. Now the operator which turns / into £' is a quadrantal versor (§ 53) ; and, as its axis is the vector 7, we may call it i. Thus ^ = h or K=iJ. (1) e/ Similarly we may put -j^=y, or 7=:yZ, (2) and '-j—hy or J—kL (3) G 1 44 QUATBBNIONa [OHAP. 11 [It may here be noticed^ merely to shew the symmetry of the system we are explaining^ that if the three mutnally per- pendicular vectors /, /, Jf be made to revolve about a line equally inclined to all, so that / is brought to coincide with /, / will then coincide with K, and K with 7: and the above equations will still hold good, only (1) will become (2), (2) will become (3), and (3) will become (1).] 67. By the results of § 50 we see that -/_ K K ^ J' i. e. a southward unit- vector bears the same ratio to an upward unit-vector that the latter does to a northward one ; and there- fore we have — / .• -— = i, or — /=»£ (4) ^ Similarly ~^^' ^^ —K=JI; (5) and —=-=:*, or ^I=lkJ. (6) 68. By (4) and (1) we have -./=ije'=:i(ic/) = i»/. Hence i» = — 1 (7) And, in the same way, (5) and (2) give >'=-!. (8) and (6) and (3) ^^^^ ^^^ Thus, as the directions of 7, e/", K are perfectly arbitrary, we see that tAe square of every quadrantal veraor is negative unity. [Though the following proof is in principle exactly the same as the foregoing, it may perhaps be of use to the student, in shewing him precisely the nature as well as the simplicity of the step we have taken. A' -OC 8BCT. 69.] PBODUCTS AND QUOTIENTS OF VECTORS. 45 Let ABjt be a semicircle^ whose centre is 0, and let OB be perpen- dicular to AOA\ Inen -=-, = q suppose^ is a qua- drantal versor, and is evidently equal 6a' to ^; §§50,53. „ ^ 62 OB 62 _ Hence q^ = -=•• ^=^ = •^=^ = — 1. 1 ^ 05 J OA -" 69. Having thus found that the squares of iyj, k are each equal to negative unity ; it only remains that we find the values of their products two and two. For, as we shall see, the result is such as to shew that the value of any other combination whfit- ever of i,j, k (as factors of a product) may be deduced from the values of these squares and products. Now it is obvious that (i. e. the versor which tiuns a westward unit- vector into an upward one will turn the upward into an eastward unit) ; or JE'=y(-/)=-y7. (10) [The negative sign, being a mere nimierical factor, is evidently commutative with^; indeed we may, if necessary, easily assure ourselves of the fact that to turn the negative (or reverse) of a vector through a right (or indeed any) angle, is the same thing as to turn the vector through that angle and then re- verse it.] Now let us operate on the two equal vectors in (10) by the same versor, i, and we have iK=.i {—jl) = —iJL But by (4) and (3) 46 QUATERNIONS. [chap. II. Comparing these equations we have -ijl=-kl: or, hy § 54 (end), ij = i, " and symmetry g^ves jh ^ i, - (11) M = J. The meaning of these important equations is very simple; and is, in fact, obvious from our construction in § 54 for the multiplication of versors; as we see by the annexed figure, where we must remember that i, J, k are quadrantal versors whose planes are at right angles, so that the figure represents a hemisphere divided into quadrantal triangles. Thus, to show that ij = i, we have, being the centre of the sphere, iV", ^, Sy F'the north, east, south, and west, and Z the zenith (as in § 65) ; jOW^OZy whence iJOW^idZ^z OS^IOW. 70. But, by the same figure, iON^OZy whence ji ON = jOZ ^OE z:z-^6W ^-kON. 7L From this it appears that ji^^ky and similarly hj — — iy ► (12) ih =— y,^ and thus, by comparing (11), jk^-kj= iy I ((11), (12)). ki ^—ik — J, SECT, 72.] PRODUCTS AND QUOTIENTS OP VECTORS. 47 These equations, along with i2=y2 = ^2=_l ((7), (8), (9)), eontain essentially the whole of Quaternions. *But it is easy to see that, for the first group, we may substitute the single equation ijk=.-\, (13) since from it, by the help of the values of the squares of i,y, k, all the other expressions may be deduced. We may consider it proved in this way, or deduce it afresh from the figure above, thus k6N= OW, jhON^ jOW^ OZy ijkm = ij6W^iOZ=:08^-^6N. 72. One most important step remains to be made, to wit the assumption referred to in § 64. We have treated i,y, k simply as quadrantal versors; and I, J, K 2iB unit- vectors at right angles to each other, and coinciding with the axes of rotation of these versors. But if we collate and compare the equations just proved we have ■ = h (11) !»./■= K, (1) \ji=-k, (12) ijJ=-K, (10) with the other similar groups symmetrically derived from them. Now the meanings we have assigned to i,/, k are quite independent of, and not inconsistent with, those assigned to 7, J, K. And it is superfluous to use two sets of characters when one will suffice. Hence it appears that i,j, k may be substituted for lyJ^K; in other words, a unit-vector when em- ployed as a factor may be considered as a quadrantal versor whose plane is perpetidicular to the vector. This is one of the main elements of the singular simplicity of the quaternion calculus. 48 QUATEBNIONS. [OHAP. IL 78. Thus tie product, and therefore the quotient, of two per- pendicular vectors is a third vector perpendicular to both. Hence the reciprocal (§ 51) of a vector is a vector wliich has. the opposite direction to that of the vector, and its length is the reciprocal of the length of the vector. The conjugate (§ 52) of a vector is simply the vector reversed. Hence, by § 52, if a be a vector {Taf = aKa = a(-a) = -a'. 74. We may now see that every versor may be represented by a power of a unit-vector. For, if a be any vector perpendicular to i (which is any definite unit-vector), ia , = ^, is a vector equal in length to a, but perpendicular to both i and a ; i^ a = — a, ^'a =— ia =— /3, i*a = — i)3 = — i»a = a. Thus, by successive applications of i, a is turned round ^ as an axis through successive right angles. Hence it is natural to defme i^ as a versor which turns any vector perpendicular to i through m right angles in the positive direction of rotation about i as an axis. Here m may have any real value whatever, for it is easily seen that analogy leads us to interpret a negative value olmas corresponding to rotation in the negative direction. 75. From this again it follows that any quaternion may be exvressed as a power of a vector. For the tensor and versor elements of the vector may be so chosen that, when raised to the same power, the one may be the tensor and the other the versor of the given quaternion. The vector must be, of course, perpendicular to the plane of the quaternion. 76. And we now see, as an immediate result of the last two SECT. 78.] PRODUCTS AND QUOTIENTS OF VECTORS. 49 sections^ that the iudex-law holds with regard to powers of a quaternion (§ 63). 77. So far as we have yet considered it, a quaternion has been regarded as the product of a tensor and a versor : we are now to consider it as a %um. The easiest method of so analysing it seems to be the following. Let -=■ represent any quaternion. Draw BC perpendicular to OA^ pro- duced if necessary. Then, § 19, Qfi = OC+CB. But, § 22, 0C=x02 where a: is a number, whose sign is the same as that of the cosine of AAOB. Also, § 73, since CB is perpendicular to OAy CB^yOA, where y is a vector perpendicular to OA and CB, i. e. to the plane of the quaternion. OB xOA + yOA Hence OA OA = a?+y. Thus a quaternion, in general, may be decomposed into the sum of two parts, one numerical, the other a vector. Hamilton calls them the scalae, and the vector, and denotes them respect-* ively by the letters S and F prefixed to the expression for the quaternion. 78. Hence q=: SqA- Vq, and if in the above example OB OA = ?. then OB^OC-^ CB^ Sq.OA+Vq. OA. 50 QUATERNIONS. [cHAP. II. [The points are inserted to shew that S and F apply only to q, and not to qOAJ] The equation above gives OC:=iSq.OA, CB = Vq.OA. 79. If, in the figure of last section, we produce BC to 2)^ so as to double its lengthy and join OL, we have by § 52^ ^^Kq^SKq^FKq; .-. 00= 0C+ CD = SKq,02 + FKq.OA, Hence 0C=8Kq.02, and CB^ FKq.OA. Comparing this value of OC with that in last section, we find SKq^Sq, (1) or the scalar of the conjugate of a quaternion is eqnal to the scalar of the quaternion, Agaiuj CB= — CB by the figure, and the substitution of their values gives FKq = --Fq, (2) or the vector of the conjugate of a quaternion is the vector of the quaternion reversed. [We may remark that the results of this section are simple consequences of the fact that the symbols S, F, K are com- mutative*. Thus SKq = KSq = 8q, since the conjugate of a number is the number itself; and FKq = KFq = ^Fq {^ Vs). * It iB curious to compare the properties of these quaternion symbols with those of the Elective Symbols of Logic, as given in Boole's wonderful treatise on the Lcbws of Tk(mght ; and to think that the same grand science of mathematical analysis, by processes remarkably similar to each other, reveals to us truths in the science of position fiur beyond the powers of the geometer, and truths of deductive reasoning to which unaided thought could never have led the lo- gician. SECT. 81.] PRODUCTS AND QUOTIENTS OP VECTOBS. 51 Again, it is obvious that iSq = 82q, iFq^ Flq, and thence ^Kq = K:Lq,'] 80. Since any vector whatever may be represented by where x, y, z are numbers (or Scalars), and i, jy k may be any three non-coplanar vectors, §§23, 25 — though they are usually understood as representing a rectangular system of unit- vectors — and since any scalar may be denoted by «? ; we may write, for any quaternion q, the expression q = w-\-xi-\-yj'\-zh (§ 78). Here we have the essential dependence on four distinct numbers, from which the quaternion derives its name, exhibited in the most simple form. And now we see at once that an equation such as where ^=w'-^'X^i-\-/j-\-zk, involves, of course, the four equations w'= w, a?'= Xy y = y, / = z. 8L We proceed to indicate another mode of proof of the distributive law of multiplication. We have already defined, or assumed (§ 61), that P y /3 + y — I — = i a a a or fia-'A-ya-'^ = {p-^y)a-'', and have thus been able to understand what is meant by adding two quaternions. But, writing a for a-^ we see that this involves the equality (i3-hy)o = ^a 4- ya; from which, by taking the conjugates of both sides, we derive a'(i8'+/) = a'^+aV (§55). H % 52 QUATERNlONa [cHAP. II. And a combination of these results (putting /3-f-y for a in the latter^ for instance) g^ves (/3-fy)(^+/) = 0+y)/3' + (i8+y)/ = ^/3' + yi3'4-/3y' + y/ by the former. Hence the distributive principle is true in the multiplication of vectors. It only remains to show that it is true as to the scalar and vector parts of a quaternion^ and then we shall easily attain the general proof. Now, if a be any scalar, a any vector, and y any quaternion, [a'\'6)q = aj' + aj. For, if /3 be the vector in which the plane of q is intersected by a plane perpendicular to a, we can find other two vectors, - y and 5, in these planes such that y ^ And, of course, a may be written — ; so that ^ y /3 y /3 = aq-\- aq. And the conjugate may be written /(«'4-a') = /a'-f/a (§ 55). Hence, generally, («-|-a)(^-fi3) = ab + afi-^-ba-^-afi: or, breaking up a and b each into the sum of two scalars, and a, fi each into the sum of two vectors, (fli -f «a -f ai -ha2)(^i -h^a -f i3i -f iQa) = K + «2)(<^i+^2) + («i-f«2)(/3i+i8,) + (*i-f*0(«i+aO + (ai-fa,)(/3i+i3, SECT. 84.] PRODUCTS AND QUOTIENTS OP VECTORS. 53 (by what precedes, all the factors on the right are distributive, so that we may easily put it in the form) + («a + aO(ia+^3). Putting «i +ai = py a^'^a^= q, h^ -f-^i = r, h^ +^a = s, we have (jo + ^)(r + *) =jor+^* 4 ?/• + ?«. 82. For variety, we shall now for a time forsake the geo- metrical mode of proof we have hitherto adopted, and deduce some of our next steps from the analytical expression for a qua- ternion given in § 80, and the properties of a rectangular system of unit- vectors as in § 7 1 . We will commence by proving the result of § 77 anew. 83. Let Then, because by § 71 every product or quotient of i^j, k is reducible to one of them or to a number, we are entitled to assume a where w, ^, r], fare numbers. This is the proposition of § 80. 84. But it may be interesting to find w, £, t;, f in terms of ^j y, Zy x\ /, /. We have or as we easily see by the expressions for the powers and products of i, j, k, given in § 7 1 . But the student must pay particular 54 QUATERNIONS. [cHAP. IT. attention to the order of the factors^ else be is certain to make mistakes. This (§ 80) resolves itself into the four equations /= <az + (^ —rix. The three last equations g^ve which determines <a. Also we have^ &om the same three^ which, combined with the first, gives y / — zy zx — xz xy — yx ' and the common value of these three fractions is then easily seen to be . x^-^y^ + z^ It is easy enough to interpret these expressions by means of ordinary coordinate geometry : but a much simpler process will be furnished by quaternions themselves in the next chapter, and, in giving it, we shall refer back to this section. 85. The associative law of multiplication is now to be proved by means of the distributive (§ 81). We leave the proof to the student. He has merely to multiply together the factors W'\-xi-{'yj'\-zk, w'+x'i-\-yj-\-sfk, and v/^ + iiif'i-\-2^'j-\'/'k, as follows : — First, multiply the third factor by the second, and then multiply the product by the firit; next, multiply the second factor by the first and employ the product to multiply the third : SECT. 88.] PRODUCTS AND QUOTIENTS OP YECTORS. 55 always remembering that the multiplier in any product is placed before the multiplicand. He will find the scalar parts and the coefficients of f, /, hy in these products, respectively equal, each to each. 86. With the same expressions for a, j3, as in section 83, we have = ^{xx'-\-yy 4- ^/) + (y^— ^/)i+ {zx'^xz)j-\- (xtf-yx") h. But we have also /3a = — (a?ar'+^/H-^/)— (y/— ^/)i-(2r/-;p/)y--(^— yarO*- The only difference is in the %ign of the vector parts. Hence ' /Sa)3 = /S/3a, (1) ra/3=-r/3a, (2) also aP+fia= 2 Safi, (3) aP-fia = 2raA (4) and, finally, by § 79, . afi=Kfia (5) 87. If a = /3 we have of course (§ 25) x = x', y=/, z = /, and the formulae of last section become which was anticipated in § 73, where we proved the formula and also, to a certain extent, in § 25. 88. Now let q and r be any quaternions, then Sqr = S.{8q+ Fq){8r^ Fr), = S.{8qSr + Sr.Fq+Sq,rr+ Fq Fr), ^SqSr + S.FqFr, since the two middle terms are vectors. 56 QUATERNIONS. [CHAP. II. Similarly, Srq = 8r8q-\- 8, Fr Vq. Hence, since by (1) of § 86 we have 8.rqrr ^ 8.rrrq, we see that Sqr=z 8rq, , (1) a formula of considerable importance. It may easily be extended to any number of quaternions, because, r being arbitrary, we may put for it rs. Thus we have (putting a dot after the 8 to shew that it refers to the whole product that follows it) 8,qrs = 8.r8q = 8.8qr by a second application of the process. In words, we have the theorem — tke scalar of the product of any number of given qua- ternions depends only upon the cyclical order in which they are arranged. 89. An important case is that of three factors, each a vector. The formula then becomes 8.afiy = /S./Sya = /S.ya/3. But 8.aPy = 8.a{8py+ F^y) =1 8,a Fpyy since a8fiy is a vector, = ^8.aFyPy by (2) of § 86, = ^8.a{8yfi+Fyfi) =i^8.ayp. Hence the scalar of the prodtcct of three vectors changes sign when the cyclical order is altered. Other curious propositions connected with this will be given later, as we wish to devote this chapter to the production of the fundamental formulae in as compact a form as possible. 90. By (4) of § 86, Hence 2 F.aF^y = F.a {fiy-yP) SECT. 92.] PRODUCTS AND QUOTIENTS OF VECTORS. 57 (by multiplying both by a, and taking the vector parts of each (by introducing the null term ^ay—^ay). That is 2r.ar)3y = jr(a/3H-)9a)y-r.()8/Say + ^ray + /S'ay./3H- Vay.ff) = r.{28aP)y-2r.fi8ay (if we notice that KFay.^^- ^.^^ay, by (2) of § 86). Hence F.aFfiy = ySafi^fiSya, (1) a formula of constant occurrence. Adding aSpy to both sides we get another most valuable formula Kafiy = aSPy-pSya + ySaP; (2) and the form of this shews that we may interchange y and a without altering the right-hand member. This gives F.afiy=:F.yfia, a formula which may be greatly extended. 91. We have also F, Fa^Fyb = - r. FybFa^ by (2) of § 86 : = bS.yFa^-yS.bFafi = b S.a^y^yS.afib, :=z^pS,aFyb-\-aS.^Fyb=^fiS.ayb-haS.fiyb, all of these being arrived at by the help of § 90 (1) and of § 89 ; and by treating alternately Fafi and Fyb as simple vectors. Equating two of these values, we have bS.afiy = aS,Pyb + pS.yab-\-yS.a^by (3) a very useful formula, expressing any vector whatever in terms of three given vectors. 92. That such an expression is possible we knew already by § 23. For variety we may seek another expression of a similar 58 QUATERNIONS. [CHAP. II. character^ by a process which differs entirely from that employed in last section. a, p, Y being any three vectors, we may derive from them three others Fafi, Ffiy, Fya\ and, as these will not generally be coplanar, any other vector h may be expressed as the sum of the three, each multiplied by some scalar (§ 23). It is re- quired to find this expression for h. Let 5 = a? Fafi ^yF^y + z Fya. Then Syb = xS.ya^ = xSafiy, the terms in y and z going out, because S.yFpy = S.yfiy = S0y^ = y^Sfi = 0, for y2 is (§ 73) a number. Similarly Sfib ^zS, Pya = zS. afiy, and SaJb = ^S.afiy. Thus bS. afiy = Fa^Syb + FfiySab + FyaSpb (4) 93. We conclude the chapter by showing (as promised in § 64) that the assumption that the product of two parallel vectors is a number, and that of two perpendicular vectors a third vector perpendicular to both, is not only useful and convenient but absolutely inevitable if our system, is to deal indifferently with all directions in space. We abridge Hamil- ton's reasoning. Suppose that there is no direction in space pre-eminent, and that the product of two vectors is something which has quantity, so as to vary in amount if the factors are changed, and to have its sign changed if that of one of them is reversed ; if the vectors be parallel, their product cannot be, in whole or in part, a vector inclined to them, for there is nothing to determine the direction in which it must lie. It cannot be a vector parallel to them ; for by changing the sign of both factors the product is unchanged, whereas, as the whole system SECT. 93.] PRODUCTS AND QUOTIENTS OF VECTORS. 59 has been reversed, the product vector ought to have been reversed. Hence it must be a number. Again, the product of two perpendicular vectors cannot be wholly or partly a number, because on inverting one of. them the sign of that number ought to change ; but inverting one of them is simply equivalent to a rotation through two right angles about the other, and (from the symmetry of space) ought to leave the number unchanged. Hence the product of two perpendicular vectors must be a vector, and an easy extension of the same reasoning shows that it must be perpendicular to each of the factors. It is easy to carry this farther, but enough has been said to show the character of the reasoning. EXAMPLES TO CHAPTER II. 1. It is obvious from the properties of polar triangles that any mode of representing versors by the sides of a triangle must have an equivalent statement in which they are represented by angles in the polar triangle. Show directly that the product of two versors represented by two angles of a spherical triangle is a third versor repre- sented by the stt;pple?/ient of the remaining angle of the triangle ; and determine the rule which connects the directions in which these angles are to be measured. 2. Hence derive another proof that we have not genei^ally pq = qp. 3. Hence show that the proof of the associative principle, § 67, may be made to depend upon the fact that if from any point of the sphere tangent arcs be drawn to a spherical conic, 12, 60 QUATERNIONS. [cfHAP. II. and also arcs to the foci, the inclination of either tangent arc to one of the focal arcs is equal to that of the other tangent arc to the other focal arc. 4. Prove the formulae 2 S.aPy = apy--yPa, 2r.a/3y= afiy+ypa. 5. Show that, whatever odd number of vectors be represented ^y tt, )3, y, &c., we have always V.afiyU = F.€bypa, F.aPybfCv = FrjCfbyfia, &c. 6. Show that STapFpyFya =-{S.aPy)\ F.FafiF^yFya= Fa^iy^SaP-SfiySya)-^ , and F.{FapF.FfiyFya) = {pSay-aSfiy)S.aPy,- 7. If a, p, y be any vectors at right angles to each other, show that (a>-f /3» + y»)/S.aj3y = a*Ffiy+fi*Fya + y*Fap. 8. If a, )3, y be non-coplanar vectors, find the relations among the six scalars, a?, y, z and f, »y, f, which are implied in the equation xa^-y^^-zy = f r^y + TyFyaH-fra^. 9. If a, )3, y be any three non-coplanar vectors, express any fourth vector, 5, as a linear function of each of the following sets of three derived vectors, F.ya^, r.a^y, F.^ya, and F.FapFpyFya, FF^yFyaFafi, FFyaFa^F^y. 10. Eliminate p from the equations Sap = <?,' S^p = b, Syp = c, Sbp = d, where a, ^, y, 5 .are vectors, and a, b, c, d scalars. CHAPTER III. INTERPEETATIONS AND TRANSFORMATIONS OF QUATERNION EXPRESSIONS. 94. 4 MONG the most useful characteristics of the Calculus ■^^ of Quaternions the ease of interpreting its formulse geometrically, and the extraordinary variety of transformations of which the simplest expressions are susceptible, deserve a prominent place. We devote this Chapter to the more simple of these, together with a few of somewhat more complex character but of constant occurrence in geometrical aid physical investigations. Others will appear in every succeeding Chapter. It is here, perhaps, that the student is likely to feel most strongly the peculiar difficulties of the new Calculus. But on that very account he should endeavour to master them, for the variety of forms which any one formula may assume, though puzzling to the beginner, is of the most extraordinary advantage to the advanced student, not alone as aiding him in the solution of complex questions, but as affi)rding an invaluable mental discipline. 95. If we refer again to the figure of § 77 we see that 0C= OBqosAOB, ' CB^OB^mAOB. ^ Hence, it 62 = a, OB = p, and Z AOB = 0, we have OB = Tfi, OA = Ta, OC = Tp cos^, CB = Tl3 sinO. 62 QUATERNIONS. [cHAP. III. „ ^p OC Tp ^ Hence 8- =. -p—- = -^ cos^. a UA Ta Similarly TF^ = -^ = ^sind?. Hence, if c be a unit- vector perpendicular to a and /3, or UOA a we have V- = -^ sin^ e. a 2a 96. In the same way we may shew that and ra)3= TaT^smBri where t^ = UVafi = - iZF- . a Thus ^>i^ *ca/tfr of the product of two vectors is the continued product of their tensors and of the cosine of the supplement of the contained angle. The tensor of the vector of the product of two vectors is the con- tinued product of their tensors and the sine of the contained angle; and the versor of the same is a unit^-vector perpendicular to both, and such that the rotation about it from the. first vector to the second is right-handed or negative. Hence TVafi is double the area of the triangle two of whose sides are a, /3. 97. (a.) In any triangle ABC we have ' AC=AB + SC. Hence AC' = 8 AC AC = 8. AC {IS -f £C). With the usual notation for a plane triangle the interpretation of this formula is SECT. 99.] INTERPRETATIONS AND TRANSFORMATIONS. 63 — i'* = —be COB A -^ab cos C, or b =z a cos C+{?cos A. (6,) Again we have, obviously, F.ABAC=:zr.AB{A£-\-BC) =:r.ABBCy or cb sin A = ca sin By , sin ^ sin B sin C whence = = . a b c These are truths, but not truisms, as we might have been led to fancy from the excessive simplicity of the process employed. 98. Prom § 96 it follows that, if a and ^ be both actual (i. e. non-evanescent) vectors, the equation 8,afi = shews that cos^ = 0, or that a is perpendicular to /3. And, in fact, we know already that the product of two perpendicular vectors is a vector. Again, if Va^^% we must have sin ^ = 0, or a u parallel to /3. We know already that the product of two parallel vectors is a scalar. Hence we see that « o r^ o ap = u is equivalent to a = Fyj3, where y is an undetermined vector ; and that is equivalent to a = xp, where x is an undetermined scalar. 99. If we write, as in § 83, a = ix +jy -^kz, /3 = ix'^-jy'^-My 64 QUATBRNlONa [chap. III. we have, at once, by § 86, Safi =: — a-y — yy'— zz' \ r r r r r r ^ where r = y/x^ +^' + 2;% / = y/» +/* + /*. Also Fa^ = „•'{ i:f-=^ i+ ^■Z^j+ ^^i}. ^ rr rr ^ rr ^ These express in Cartesian coordinates the propositions we have just proved. In commencing the subject it may perhaps assist the student to see these more famiUar forms for the quaternion expressions ; and he will doubtless be induced by their appear- ance to prosecute the subject, since he cannot fail even at this stage to see how much more simple the quaternion expressions are than those to which he has been accustomed. 100. The expression « ^ may be written S.{Va^)y because the quaternion aj3y may be broken up into {8ap)y + {raP)y of which the first term is a vector. But, by § 96, 5.(ra/3)y = TaT^QmeSriy. Here Trj =^ l, let <t> be the angle between ri and y, then finally S.afiy --TaTfiTy sin cos <^. But as ly is perpendicular to a and /3, Ty cos <[> is the length of the perpendicular from the extremity of y upon the plane of a, p. And as the product of the other three factors is (§ 96) the area of the parallelogram two of whose sides are a, j8, we see that the magnitude of S.afiy, independent of its sign, is tAe volume of the parallelepiped of which three coordinate edges are a, )3, y; or six times the volume of the pyramid which has a, )3, y for edges. SECT. 102.] INTERPRETATIONS AND TRANSFORMATIONS. 65 IQL Hence the equation S.a^y = 0, if we suppose a, ^, y to be actual vectors, shews either that sin fl = 0, or cos </) = 0, i. e. two of the three vectors are jparallel, or all three lie in one plane. This is consistent with previous results, for if y = pfi we have S.aPy=p8.aP^ = 0; and, if y be coplanar with a, p, we have y ^pa-^qfi, and S.apy = S.aPipa-^qP) = O'. 102. This property of the expression S.a^y prepares us to find that it is a determinant. And, in &ct, if we take a, ^ as in § 83, and in addition y = iy'+y/'+*/', we have at once X y z x' / / The determinant changes sign if we make any two rows change places. This is the proposition we met with before (§ 89) in the form S.Q^y = -^8. Pay = S.fiya, Sec. If we take three new vectors Oi = ix-^jV-^h/^y yi =i^-|-y/-|-*/', we thus see that they are coplanar if a, /3, y are so. That is, if S.afiy = 0, then &aj/3iyj = 0. K 66 QUATKBNIOKS. [oBap. IE 103. We hsve, bj § 52. (Tqy « qKq = (8i+Fq)(8q-Fq) (§ 79), = (Sqy-iFqy by algebra, = {8qy+(TFqy (§ 73). If ^ = afij we have Kq = pa, and the formula becomes ap.fia = a«/3« = (Saj8)«-(rai3)». In Cartesian coordinates this, is More generally we have = jf Zf JTj (§ 56) = (r?)' (2V)» (§ 62). If we write jf = «? +a = w +ip +^y +fe, this becomes (w^+x^ +y» + -2;»)(«?'» +a:'» +/« -h/^) -h («?/ -h «? y -h ;Ka?' — or/)* + («?/ -h «p'2J + ir/ — ^/)', a formula of algebra due to Euler. 104« We have^ of course, by multiplication, (a+/3)« = a» + ai9-l-/3a + 0» = a»-h2Sa/3 + 0» (§86(3)). Translating into the usual notation of plane trigonometry, this becomes c» = a» - 2 a* cos C+ 5*, the common formula. Again, ir(a+/3)(a-i8) = - ra)3+ V^a = -2 Fa/S (§ 86 (2)). Taking tensors of both sides we have the theoren\, tie paral- lelogram wAose sides are parallel and equal to the diagonals of a given parallelogram, has dovile its area (§ 96). SECT* 106.] INTERPEETATIOKS AND TRANSFOEMATIONS. 67 Also iS(a+j3)(a-/3) = o»-/3% and vanishes only when a^ = /3», or Ta^ Tfi; that is, Ue diagonals of a parallelogram are at right angles to one another y when, and only wheUy it is a rhombus. Later it will be shewn that this contains a proof that the angle in a semicircle is a right angle. 105. The expression p = aj9a~* obviously denotes a vector whose tensor is equal to that of )3. But we have ^.^,^ = 0, so that p is in the plane of a, )3. Also we have Sap = Saft so that )3 and p make equal angles with a, evidently on opposite sides of it. Thus if a be the perpendicular to a reflecting surface and /3 the path of an incident ray, p will be the path of the reflected ray. Another mode of obtaining these results is to expand the above expression, thus, § 90 (2) p = 2a-*/Sa)3-i9 = a-H^ai3-ira/3), so that in the figure of § 77 we see that if OA = a, and OB = /3, we have OB = p = a/3a"^ 106. For any three coplanar vectors the expression p = a^y is (§ 101) a vector. It is interesting to determine what this vector is. The reader will easily see that if a circle be described about the triangle, two of whose sides are (in order) a and )3, and if from the extremity of j3 a line parallel to y be drawn K % 68 QUATBENIONS. [CHAP. III. again cutting the circle^ the vector joining the point of inter- section with the origin of a is the direction of the vector apy. For we may write it in the form p = a^«/3- V = -{Tpy ap-^y = -(I]3)» | y, which shows that the veraor which turns fi into a direction parallel to a, turns y into a direction parallel to p. And this is the long known property of opposite angles of a quadrilateral inscribed in a circle. Hence if a, /3, y be the sides of a triangle taken in order, the tangents to the circumscribing circle at the angles of the triangle are parallel respectively to a/3y, Pya, and yafi. Suppose two of these to be parallel, i. e. let aPy=xPya=xayP (§ 90), since the expression is a vector. Hence /3y=a7/3, which requires either a?=l, ry^=0, or y||/3, a case not contemplated in the problem ; or a? = — 1, SPy=0, i. e. the triangle is right-angled. And geometry shews us at once that this is correct. Again, if the triangle be isosceles, the tangent at the vertex is parallel to the base. Here we have a?^=a/3y, or x{a -h y) = a{a -h y)y ; whence ir=y" = a^, or Ty=:Ta, as required. As an elegant extension of this proposition the reader may prove that the vector of the continued product afiyb of the vector-sides of a quadrilateral inscribed in a sphere is parallel to the radius drawn to the comer (a, 6). SECT. 107.] INTERPRETATIONS AND TRANSFORMATIONS. 69 107. To exemplify the variety of possible transformations even of simple expressions, we will take two cases which are of frequent occurrence in applications to geometry. Thus TO) + a)=T(p-a), [which expresses that if Q2=a, 6jL--a, and QP=p, we have AP^A'P and thus that P is any point equidistant from two fixed points,] may be written (p + a)«=(p-a)% or p« + 2/Sap + a> = p« — 2iSap + a» (§ 104), whence Sap=0. This may be changed to ap-h pa=0, or ap + Jrap=0, a or finally, TVU^=zl, a aU of which express properties of a plane. Again, Tp-Ta may be written ^p _ a (p-ha)»-2/Sa(p + a) = 0, p = (p-i-a)-*a(p-|-a), '8'(p + a)(p— a)=0, or finally, 7.(p + a)(p-a) = 2Trap. All of these express properties of a sphere. They will be interpreted when we come to geometrical applications. 70 QUATERNIONS. [cHAP. HI. 108. We have seen in § 95 that a quaternion may be divided into its scalar and vector parts as follows : — ^ = 5^+ F^ = -^ (cosO + esin^r; a a a la where is the angle between the directions of a and p, and a € = UV- is the unit-vector perpendicular to the plane of a a and j3 so situated that positive (i. e. left-handed) rotation about it turns a towards /3. Similarly we have (§ 96) ap = 8afi+Vap = Ta2)3 (— cos fl— € sin 0), $ and € having the same signification as before. 109. Hence, considering the versor parts alone, we have £/■- = — Uafi = cos ^ + € sin ^. a Similarly t/^ = cos</)-f €sin^; </> being the positive angle between the directions of y and J3, and € the same vector as before, if a, ^, y be coplanar. Also we have U- = cos (^ + <^) + c sin {0+<t)). a But we have always ^ . - = ~ , and therefore pa a' /3 a a or cos(0+^)-f €sin(<^4-^) = (cos^ + €sin<^)(cos^+esin^) = cos ^ cos fl— sin (^ sin ^ + € (sin </> cos d+ cos </> sin 0), from which we have at once the fundamental formulae for the SECT. 111.] INTERPRETATIONS AND TRANSFORMATIONS. 71 cosine and sine of the sum of two arcs^ hj equating separately the scalar and vector parts of these quaternions. And we see^ as an immediate consequence of the expressions above, that cos m$'^€ sin mO = (cos 0+ c sin d)** if w be a positive whole number. For the left-hand side is a versor which turns through the angle mO 2it once, while the right-hand side is a versor which effects the same object by m successive turnings each through an angle 6. See § 8. UO. To extend this proposition to fractional indices we have only to write - for 6, when we obtain the results as in ordinary trigonometry. Prom De Moivre's Theorem, thus proved, we may of course deduce the rest of Analytical Trigonometry. And as we have already deduced, as interpretations of self-evident quaternion transformations (§§ 97, 104), the fundamental formulsB for the solution of plane triangles, we will now pass to the consideration of spherical trigonometry, a subject specially adapted for treat- ment by quaternions ; but to which we cannot afford more than a very few sections. The reader is referred to Hamilton's works for the treatment of this subject by quaternion exponentials. HI. Let a, /3, y be unit- vectors drawn from the centre to the comers A, -B, C of a triangle on the unit-sphere. Then it is evident that, with the usual notation, we have (§ 96), Sap ss —cos c, Sfiy = —cos a, 8ya = —COS dy TVafi= sine?, TVpy ^ sin a, TVya= smb. Also UVafi^ UVfiy, UVya are evidently the vectors of the corners of the polar triangle. Hence S. UFap UVpy = cos £, &c., TV. UVap UVfiy = sin B, &c. 72 QUATERNIONS. [chAe. IIL Now (§ 90 (1)) we have SVapVpy = S.aV.prpy Remembering that we have SFaprpy = TFapTVPySMFapUrpy, we see that the formula just written is equivalent to sinasin^roosJSs:— cosaco8(?+cosdj or co8& = co8acos(?+sina8in(?co8^. 112« Again^ F.rapFPy--p8afiy which gives TF.FapF^ = S.aPy = S.aFPy = S.^Fya = S.yVafi, or sinasint^sinJS = sinasinj?^ s sin^sinjo^ = sint^sinj^^; where j9a is the arc drawn from A perpendicular to BC, &c. Hence ^^fia = sin <; sin B, sinasinc . _, sin 19b = — ' — 2 — sm /!• smp^ = sin a sin B. 113. Combining the results of the last two sections^ we have Fafi. Fpy = sin a sin <; cos £ — j3 sin a sin csiaB = sin a sin <; (cos JS^ j3 sin JS). Hence U. FaB Fpy = (cos J8 - j8 sin B), ] and U.FyPFpa = (cos B+fimn B). These are therefore versors which turn the system negatively or positively about OB through the angle B. As another instance^ we have ■\ ^-^ =&c Say-^SafiS^' SECT. 114.] INTEKPBBTATIONS AND TBANSFORMATIONS. 73 The interpretation of each of these forms gives a different theorem in spherical trigonometry. 114. A curious proposition, due to Hamilton, gives us a qua- ternion expression for the spherical excess in any triangle. The following proof, which is very nearly the same as one of his, though by no means the simplest that can be given, is chosen here because it incidentally gives a good deal of other in- formation. We leave the quaternion proof as an exercise. Let the unit-vectors drawn from the centre of the sphere to Ay By C, respectively, be a, /3, y. It is required to express, as an arc and as an angle on the sphere, the quaternion The figure represents an orthographic projection made on a plane perpendicular to y. Hence C is the centre of the circle BEe. Let the great circle through Ay B meet BEe in Ey Cy and let BE be a quadrant. Thus BE represents y (§ 72). Also make EF=uiB== pa-\ Then, evidently, BF=fia-'yy which gives the arcual representation required. 74 QUATBKNIONS. [cHAP.IIL Let BF cut Ee'm 0. Make Ca =r EQj and join B^ a, and a, F. Obviously^ as 2> is the pole of Ee, fiaiBBi quadrant ; and since FO = Ca, Ga ^ FC, a quadrant also. Hence a is the pole of DO, and therefore the quaternion may be represented by the angle DaF. Make Cb^ Ca, and draw the arcs Pafi, Pha from P, the pole of AB. Comparing the triangles Fba and ea^, we see that Fa = e^. But, since P is the pole of AB, F^a is a right angle : and therefore as Pa is a quadrant, so is F^. Thus AB is the complement of Fa or fie,, and therefore aP=^2AB. Join bA and produce it to c so that Ac =^bA; join c, P, cutting AB in o. Also join c, B, and j5, a. Since P is the pole of AB, the angles at o are right angles ; and therefore, by the equal triangles baA, coA, we have aA = Ao. But ai3 = 2AB, whenise oB = Bfi, and therefore the triangles coB and JSa^ are equal, and e, B, a lie on the same great circle. Produce cA and cB to meet in H (on the opposite side of the sphere). H and c are diametrically opposite, and therefore cP, produced, passes through H. Now Pa = Pb = PR, for they di£Per from quadrants by the equal arcs ap, ba, oc. Hence these arcs divide the triangle Eab into three isosceles triangles. But LPEb^LPHa = LaHb = Lbca. Also L Pab = ff- Lcab-L PaH, Z Pba = LPab = tt-Z cba-LPbH. Adding, 2 ZPai = It:—- Lcab-- Lcba — Lbca = -JT— (spherical excess of adc). SECT. 116.] INTBEPBBTATlOlSrS AND TEANSP0BMATI0N8. 76 But^ as Z Fa^ and Z Dae are right angles^ we have angleof/3o~*y = LFaJD^ L^aer=, LPab It = - — ^ (spherical excess oiabc). [Numerous singular geometrical theorems, easily proved ab initio by quaternions, follow from this : e. g. The arc AB, which bisects two sides of a spherical triangle abc^ intersects the base at the distance of a quadrant from its middle point. All spherical triangles, with a common side, and having their other sides bisected by the same great circle (i. e. having their vertices in a small circle parallel to this great circle) have equal areas, &c., &c.] 115. Let 0« = a', Oi = /S', 0^ = /, atid we have = Ca.BA =: JEG.F^ =^ f^. But FG is the complement of DF, Hence the an^le of the quaternion ($)%nii is half the spherical excess of the triangle whose angula/r points are at the extremities of the unit^vectors o', ^, y\ [In seeking a purely quaternion proof of the preceding pro- positions, the student may commence by showing that for any three unit- vectors we have a ^ y ^ '* The angle of the first of these quaternions can be easily assigned ; and the equation shows how to find that of fia^^y. But a still simpler method of proof is easily derived from the composition of rotations.] L 2 76 QUATERNIONS. [chap. IH. 118. A scalar equation in p^ the vector of an undetermined pointy is generally the equation of a surf ace; since we may substitute for p the expression p:=^xay where or is an unknown scalar^ and a any assumed unit- vector. The result is an equation to determine x. Thus one or more points are found on the vector xa whose coordinates satisfy the equation ; and the locus is a surface whose degree is determined by that of the equation which gives the values of x. But .a vector equation in p^ as we have seen^ generally leads to three scalar equations^ from which the three rectangular or other components of the sought vector are to be derived. Such a vector equation^ then^ usually belongs to a definite number of jmnts in space. But in certain cases these may form a line, and even a surface, the vector equation losing as it were one or two of the three scalar equations to which it is usually equivalent. Thus while the equation ap=/3 gives at once which is the vector of a definite point (since we have evidently i8a/3=0); the closely allied equation rap=/3 is easily seen to involve and to be satisfied by whatever be x. Hence the vector of any point whatever in the line drawn parallel to a from the extremity of a"*^ satisfies the given equation. 117. Again, FapTp^^iFafiy is equivalent to but two scalar equations. For it shews that SECT. 119.] INTEBPEETATIONS AND TRANSPOEMATIONS. 77 Vap and Ffip are parallel^ i. e. p lies in the same plane as a and p, and can therefore be written (§ 24) where x and y are scalars as yet undetermined. We have now Fop =yrap, rpp-xVapy which, by the given equation, lead to dy=l, or y=-, or finally X 1 ^ X which (§ 40) is the equation of a hyperbola whose asymptotes are in the directions of a and ft. US. Again, the equation r.Faprap-o, though apparently equivalent to three scalar equations, is really equivalent to one only. In fact we see by § 91 that it may be written — ajS.aj9p=0, whence, if a be not zero, we have S.aPp=0, and thus (§ 101) the only condition is that p is coplanar with a, p. Hence the equation represents the plane in which a and j9lie. lift Some very curious results are obtained when we extend these processes of interpretation to functions of a quaternion instead of functions of a mere veetor p. A scalar equation containing such a quaternion, along with quaternion constants, gives, as in last section, the equation of 78 QUATBENIONS. [CHAF. Ill a euiface, if we assign a definite value to «. Hence for succes- sive values of », we have successive surfaces belonging to a system ; and thus when o is indeterminate the equation repre- sents not a surface, as before^ but a volume, in the sense that the vector of any point within that volume satisfies the equation. Thus the equation (TV—a^ or ©•— p*=a*, or (TpY = a»-a)», represents^ for any assigned value of q^ not greater than a, a sphere whose radius is \/a*— «•. Hence the equation is satisfied by the vector of any point whatever in the volume of a sphere of radius a, whose centre is origin. AgaiUj by the same kind of investigation^ where ;=:a> -hp is easily seen to represent the volume of a sphere of radius a described about the extremity of /3 as centre. Also 8{q*)-=z —a' is the equation of infinite space less the space contained in a sphere of radius a about the origin. Similar consequences as to the interpretation of vector equa- tions in quaternions may be readily deduced by the reader. 120. The following transformation is enunciated by Hamilton {Lectures, p. 587, and JElements, p. 299). r-^(r*j»)*^» = U{rq+KrKq). Let r-*(r» j»)*y-» = t, then ^ = Ij and therefore Kt = ^* ; But (r»?»)* = rtq, or r^q^ = rtqrtq, or rq^tqrt,. Hence KqKr = tr^KrKqt-\ or KrKq = tKqKrt. SECfP..121.] INTEEPBETATIONS AND TRANSFORMATIONS. 79 Thus we have U{rq±KrKq) = tU{qr±KqKr)t, or, if we put « = V'{qr±KqKr), Ks = ±tst. Hence sKs = (Ts)' = 1 = ± stst, which, if we take the positive sign, requires *^ = ± 1, or t=: ±8-^ = ± UKs, which is the required transformation. [It is to be noticed that there are other results which might have been arrived at by using the negative sign above ; some involving an arbitrary unit-vector, others involving the ima- ginary of ordinary algebra.] 12L As a final example, we take a transformation of Hamil- ton's, of great importance in the theory of surfaces of the second order. Transform the expression in which a, )3, y are any three mutually rectangular vectors, into the form . y(ip+pK) x' which involves only two vector-constants, *, k. {T(ip+pK)}» = (ip+p^)(pi + ^p) (§§52,55) = (l* + K*)p' H- {l^pKp + p#Cpi) = (i» + ic«)p» + 2&ipicp = (i-jc)V + 4&p%. ^ Hence (^ap)« + (5^p)- + (%)» = i^^ But a'^iSapy+fi-^iSM'+y-'iSypy = p' (§§ 25, 73). 80 QUATEBNIONfl. [chap. III. Multiply by /3' and subtract^ we get (■-?)(M--(^-0(%)-={(lH!?-^-)p-+«(^.- The left side breaks up into two real factors if J3* be intermediate in value to a* and y* : and that the right side may do so the term in p* must vanish. This condition gives P* = . ,_ ,x, ; and the identity becomes 5(av/(i-^) + yv^(7-i)M(«v/(i-|^)-y^(f!-i))p Hence we must have -^=^(av'(l-^) + yv/(^4-l)> -j_j = -(av/(i-^)-yv(^-i)> where jt7 is an undetermined scalar. To determine JD, substitute in the expression for p*, and we find = (i'*+i)(«*-y')-2(«*+y')+4/3'. I' Thus the transformation succeeds if 1 / a^ which gives jd -f - = + 2 v/— JD "" a'— y* JO ~ " a" — y' Hence ^^^ = (^-i*') («'->') = ±4x/^', or (k»-i»)-»= +ya2V- SECT. 123.] INTERPBBTATIONS AND TBANSPORMATIONS. 81 Again, p= /-T— ^. > -= y - ^ and therefore Thus we have proved the possibility of the transformation, and determined the transforming vectors t, k. 122. By differentiating the equation (&V>)* + (5^P)«+(W = (^±^)' we obtain, as will be seen in Chapter IV, the following, "where p' also may be any vector whatever. This is another very important formula of transformation; and it will be a good exercise for the student to prove its truth "by processes analogous to those in last section. We may merely observe, what indeed is obvious, that by putting p^= p it be- comes the formula of last section. And we see that we may write, with the recent values of i and k in terms of o, j3, y, the identity aSap + pSpp + ySyp = ^ (^Jj.^y ^ {L-Kyp + 2(lSKp+KSip) 123. In various quaternion investigations, especially in such as involve imaginary intersections of curves and surfaces, the old imaginary of algebra of course appears. But it is to be par- ticularly noticed that this expression is analogous to a scalar and not to a vector, and that like real scalars it is commutative in multiplication with all other factors. Thus it appears, by the 82^^ QUATBRNI0N8. [cHAP. IE. same proof as in algebra, that any quaternion expression which contains this imaginary can always be broken up into the sum of two parts, one real, the other multiplied by the first power of \/— 1. Such an expression, viz. where / and /' are real quaternions, is called a biquatesnion. Some little care is requisite in the management of these ex- pressions, but there is no new diflSculty. The points to be observed are : first, that any biquatemion can be divided into a real and imaginary part, the latter being the product of '/—I by a real quaternion; second, that this \/— 1 is com- mutative with all other quantities in multiplication; third, that if two biquatemions be equal, as we have, as in algebra, 80 that an equation between biquatemions involves in general eight equations between scalars. Compare § 80. 124. We have, obviously, since \/— 1 is a scalar. Hence (§ 103) The only remark which need be made on such formulse is this, that the temor of a Hquatemion may vanish mhih both of the component quaternions are finite. Thus, if T/= Tf, and 84 K4'- 0, SECT. 125.] INTERPRETATIONS AND TRANSFORMATIONS. 83 the above formula gives The condition S.^Kf- may be written o4 Kf^^'^a, or /'=-aZ/-»=- where a is an indeterminate vector. Hence T^=Tf:=TKf=~y and therefore is the general form of a biquaternion whose tensor is zero. • 125. More generally we have, q, r, /, / being any four real and non-evanescent quaternions. That this product may vanish we must have qr = //, and q/=i^^r. Eliminating / we have q^-^qr^-^r, which gives (/"*?)* = ~ 1^ i. e. J = ^a where a is some unit- vector. And the two equations now agree in giving — r = a/, so that we have the biquaternion factors in the form ^{a-{- sT^) and -(a- V^)/ ; and their product is -^(a+ y^) (a- V^)/, which, of course, vanishes. [A somewhat simpler investigation of the same proposition may be obtained by writing the biquatemions as M % 84 QUATBRNIOKa [chap, ni- or /(/'+yZT) and (/'+>/^)/, and showing that j^'=: -/'=o, where Ta = 1.] From this it appears that if the product of two Hvectors p+o-^/— 1 and p'+ </\/^— 1 is zero^ we must have where a may be any vector whatever. But this result is still more easily obtained by means of a direct process. 126. It may be well to observe here (as we intend to avail ourselves of it in the succeeding Chapters) that certain ab- breviated forms of expression may be used when they are not liable to confose^ or lead to error. Thus we may write T^q for {TqY, just as we write cos*^ for (cos^)S although the true meanings of these expressions are. T{Ta) and cos (cos ^). The former is justifiable, as T{Ta) = Ta, and therefore T^a is not required to signify the second tensor (or tensor of the tensor) of a. But the trigonometrical usage is quite inde- fensible. Similarly we may write S'q for {SqY, &c. but it may be advisable not to use Sq^ as the equivalent of either of those just written ; inasmuch as it might be confounded with the (generally) different quantity 8.q^ or S{q% although this is rarely written without the point or the brackets. SECT. 127.] IITPBRPBBTATIONS AND TRANSFORMATIONS. 85 127. The beginner may expect to be a little puzzled with this aspect of the notation at first; but, as he learns more of the subject, he will soon see clearly the distinction between such an expression as where we may omit at pleasure either the point or the first V without altering the value, and the very diflferent one which admits of no such chatiges, without altering its value. All these simplifications of notation are, in fact, merely ex- amples of the transformations of quaternion expressions to which part of this Chapter has been devoted. Thus, to take a very simple example, we easily see that S.FapFpy = SFa^py = S.apFfiy = SaF.^Ffiy^ ^SaF{Ffiy)p = 8aF{Fyp)P = 8.aF{yp)P = 8.F{yfi)fia = SFyfiFfia = S.ypFpa z= Sec, Sec. The above group does not nearly exhaust the list of even the simpler ways of expressing the given quantity. We recommend it to the careful study of the reader. EXAMPLES TO CHAPTER III. 1. Investigate, by quaternions, the requisite formulae for changing jfrom any set of coordinate axes to another; and derive from your general result, and also from special investi- gations, the common expressions for the following cases : — (a.) Rectangular axes turned about z through any angle. 86 QUATEBKIONS. [CHAP. HI. (b.) Bectangolar axes turned into any new position by rotation about a line equally inclined to the three. (c.) Bectangnlar turned to oblique, one of the new axes lying in each of the former coordinate planes. 2. If Tp = It = 2)3= 1, and S.app = 0, show by direct trans- formations that Interpret this theorem geometrically. 3. If So^ = 0, 7a = 7/3 = 1, show that (1 +o*)j3 = 2cos ^<?p = 2 SJ.a^fi. V 4. Put in its simplest form the equation pS.raprpyVya = aF.ryaFaP + br.rafiFfiy + cr.r^rya; and show that a ^ o. a = S.pyp, &c. 1/^. Prove the following theorems, and exhibit them as pro- perties of determinants : — (a.) &(a + i8)(/3H-y)(y + o) = 2&a^y. («.) 8.Faprpyrya = -{8.aPyy. {c.) 8.F{a + P)(fi + y)F{fi-^y)(y + a)F{y + a){a + P)= -4(&a^y)^ {d.) S.F{FafiFPy)F{FfiyFya)F{FyaFaP)=-{8.afiyy. {e.) &8€f=-16(iS.a)3y)*, where j= r(r(a+i3)(/34-y)r(^ + y)(y+a)), €= r(r(i3 + y)(y + a)r(y-ha)(a + /3)), f = r(r(y + a)(a+^)r(a + /3)0+y)). 6. Prove the common formula for the product of two de- terminants'^of the third order in the form S.apy8.aiPiyi = ScMi Sfiai 8yai Sap, SfiP, SyP, Say, Spy, 8yy, CHAB, III.] EXAMPLES TO CHAPTER III. 8T y 7. The lines bisecting pairs of opposite sides of a quadrilateral are perpendicular to each other when the diagonals of the qua- drilateral are equal. \^ •^0 i. Show that (i.) 8.(f = S'qS SqT^ Fq, -— '(rf.) 8{F.apyF.fiyaF.yaP) = ^Sa^SpySyaS^afiy, (e.) F.q':=z{3 8'q-^T^Fq)Fq, (/.) qUFr^ ^Sq.UFq^TFq; and interpret each* as a formula in plane or spherical trigo- nometry. >' 9. If 9 be an undetermined quaternion^ what loci are repre- sented by {a.) {qa-'y^^a\ {b.) {qa-'y = a*, where a is any given scalar and a any given vector? y 9Z Sf^ If J be any quaternion^ show that the equation <2« = q' is satisfied, not alone by Q^±q but also, by Q = ± ^sf:^{8q.T}Fq-TFq\ (Hamilton, Lectures, p. 673.) ,2 ^1 1- Wherein consists the difference between the two equations T^^^l, and (^'=-1? What is the full interpretation of each, a being a given, and p an undetermined, vector ? 88 QUATBBNIOKS. [cHAP. HI. )U y 1 2. Knd the fall oonseqaenoes of each of the following groups of equations, both as regards the unknown vector p and the given vectors a, fi, y. SaSa^O Sap ^0, Sap =0, o.pyp^v, g^^ _^^ S.aPYp=0. , '', i 13. From §§ 74^ 109^ show that^ if c be any unit-vector, and many scalar, €** = cos— — hcsin-— • 2 2 Hence show that if a, j3, y be radii drawn to the comers of a triangle on the unit-sphere, whose spherical excess is m, right »°8rles, a+^ y+a ^+y ^ ^ /3 + y'a+i3'y + a Also that, i£Ag£, Che the angles of the triangle, we have tC tB iA y»j3»o»=;— 1. I i/14. Show that for any three vectors a, 0, y, we have {Uapy + {Ufiyy-{-{Uayy + {U.aPyy + 4: Uay.8UapSUfiy =-^2. (Hamilton, Elements, p. 388.) ^"^ 1/16. If «!, a„ a„ X, be any four scalars, and pi, p„ p, any three vectors, show that (5.piP>p.)* + (2.«,rp,p.)«+a?«(2rp,p3)»-a?'(2.«,(p,~Pi))« +2n(a?»+5pip,+tf a) = 2n(a?«+v)+2na» /^ + 2{(a:«+a,«+p,«)((rp.p,)« + 2a,«3(^« + %p.)--;p«(p.-p,)*)}; where Ua* = «i«a,«a,«. Verify this formula by a simple process in the particular case ^1 = A, =r a, ST 07 = 0. [Ibid.) CHAPTER IV. DIFFERENTIATION OF QUATERNIONS. 128. TN Chapter I we have already considered as a special -■- case the differentiation of a vector function of a scalar independent variable : and it is easy to see at once that a similar process is applicable to a quaternion ftmction of a scalar independent variable. The differential, or differential coefficient, thus found, is in general another function of the same scalar variable ; and can therefore be differentiated anew by a second, third, &c. application of the same process. And precisely similar remarks apply to partial differentiation of a quaternion function of any number of scalar independent vari- ables. In fact, this process is identical with ordinary differ- entiatio;n. 129. But when we come to differentiate a function of a vector, or of a quaternion, some caution is requisite; there is, in general, nothing which can be called a differential coefficient; and in fact we require (as already hinted in § 36) to employ a definition of a differential, somewhat different from the or- dinary one, but coinciding with it when applied to functions of mere scalar variables. 130. If r = F{q) be a ftinction of a quaternion q, dr=dFq=J^^n{F{q-\- ^)-F(q)h 00 QUATERNIONS. [cHAP. IV. where n is a scalar which is ultimately to be made infinite^ is defined to be the differential of r or Fq, Here dq maj be any quaternion whatever, and the right-hand member may be written /(?. ^)> where /is a new function^ depending on the form oiF; homo- geneous and of the first degree in ^; but not^ in general^ capable of being put in the form i{q)dq. 13L To make more clear these last remarks, we may observe that the function ^, , x thus derived as the differential oiF{q)y is distributive with respect to dq. That is Aq,r-\-s)=/{q,r)-^/iq,s), r and s being any quaternions. For Aq.r+») = J^n{F{q+'^-F{q)) =Aq>r)+Aq,»). And, as a particular case, it is obvious that if ;r be any scalar /(?, ar) = x/{q, r). 132. And if we define in the same way dF(q, r,» ) as being the value of •C.«{^(?+ ?.»•+?,*+$. )-nq,r,s, )}, where q,r,s, ,.. dq,dr,dsy are any quaternions whatever ; we shall obviously arrive at a result which may be written Aq>r,s,..,dq,dr,ds, ), SECT. 133.] DIFFERENTIATION. 91 where /is homogeneous and linear in the system of quaternions dqydrydSj and distributive with respect to each of them. Thus, in differentiating any power, product, &c. of one or more quaternions, each factor is to be differentiated as if it alone wei:^ variable ; and the terms corresponding to these are to be added for the complete differential. This differs from the ordinary process of scalar differentiation solely in the feet that, on account of the non-commutative property of quaternion mul- tiplication, each factor must be differentiated in situ. Thus d{qr) = dq.r-\-qdrf but not generally = rdq-^qdr. 133. As Examples we take chiefly those which lead to results of constant use to us in succeeding Chapters. Some of the work will be given at full length as an exercise in quaternion transformations. (1) (2'p)'=-p'. The differential of the left-hand side is simply, since 2}> is a «^^> 2TpdTp. ThatofpMs = <.«(!*^/'+^)<5'<") ^n n = 2 Spdp. Hence TpdTp = —Spdp, or dTp = -8.Updp=8 * , or dTp dp Tp^\- (2) Again, p^TpUp dp = dTp.Up'^TpdUp, N 2, 92 QUATERNIONS. [cHAP. IV. , dp dTp dVp whence -T. = 4. p Tp Up Hence dUp _ jr rfp This may be transfonned into jrdpp ^ &c. (3) {Tqy^^qKq 2TqdTi = d(qKq) = J^^n{{q^.^)K{q-^^)-qKq^, = qKdq-^dqKq, ^qKdq+K{qKdql =:2S.qKdq = 28.Kqdq. Hence rf^y = 8. UKqdq = S. I7y-»^ since Tq = My, and UKq = ?7f-^ If y = p, a vector, Kq=^Kp:=z-^p, and the formula becomes dTp^^S.Updp asin(l). But dq = Ty^E/y + f/y(?ry, , . , . dq dTq dUq which gives !^=^+-^; whence, as S — = ~^ , q Tq we have F— * = -fj^ • q Uq SECT. 133.] DIFFERENTIATION. 93 (4) ^(?')=c^((2+fy-?o = qdq + dqq = 2S.qdq-^2Sq.V€lq + 2Sdq.Fq. If ^ be a vector, os p, Sq and 8dq vanish, and we have d(p^) = 2Spdp as in (1). (5) Let q = /•*. This gives dr^ = dq. But dr = ^(y*) = qdq-^-dqq. This, multiplied Jy j and into Kq, gives ydr = q^dq-i-qdqq, and drKq:=r dqTq^-^qdqKq. Adding, we have qdr + (^rJCy = {q^ + Tj' + 2 Sjr. jr)rf^ ; whence dq, i. e. d/*, is at once found in terms of dr. This pro- cess is given by Hamilton, Lectures, p. 628. (6) qq-^ = 1, qdq-^-^dqq-^ = 0; .'. dq-^ =^^^dqq-^. If J is a vector, = p suppose, dp~^ =z—p-^dpp~^ _^_ 2^^ p^ p p ^p p^p (7) q = Sq+rq, dq=id8q-\-dFq. But dq=: Sdq-i-Fdq. 94 QUATERNIONS. [chap. IV. Comparing, we have dSq = 8dq, dVq = Vdq. Since Kq=: 8q^ Fq, we find by a similar process dKq = Kdq. 134. Saccessive differentiation of course presents no new difficulty. Thus, we have seen that d{q*) zzzdqq-^-qdq. Differentiating again, we have d'{q') = d'q.q-i-2{dqy-^qd'q, and so on for higher orders. ISqhesL vector, as p, we have, § 133 (1), d{p^) = 28pdp. Hence d*{p*) = 2{dpy + 28pd*pj and so on. Similarly d'Up = ^d{^Vpdp). Hence and d. Vpdp = V.pd'p. Bat rf-L = _i^ = £M', -=^^{{Vpdpy^p-Vpd-p-^2Vpdp8pdp). [This may be farther simplified; but it may be well to caution the student that we cannot, for such a purpose, write the above expression as ^ V.p {dpVpdp^d-pp- + 2dpSpdp}.} 135. If the first differential of q be considered as a constant quaternion^ we have, of course, d^q = 0, d^q = 0, &c., and the preceding formulae become considerably simplified. SBGT. 137.] DIFFERENTIATION. 95 Hamilton has shown that in this case Taylor^s Theorem admits of an easy extension to quaternions. That is^ we may write Aq+xdq) =/iq)+sd/(q)+ ^d^Aq) + if d*q=0; subject, of course, to particular exceptions and limita- tions as in the ordinary applications to fiinctions of scalar variables. Thus, let /(q) = j', and we have d^iq) = 2dqqdq-^2q{dqY + 2{dqyq, and it is easy to verify by multiplication that we have rigorously j'\-xdqy =q* + x{q^dq + qdqq + dqq^)+x^{dqqdq-^q{dqy-^{dqyq) + x^{dqy I which is the value given by the application of the above form of Taylor's Theorem. As we shall not have occasion to employ this theorem, and as the demonstrations which have been found are all too la- borious for an elementary treatise, we refer the reader to Hamil- ton's works, where he will find several of them. 136. To diflferentiate a function of a function of a quaternion we proceed as with scalar variables, attending to the peculiarities already pointed out. 137. A case of considerable importance in geometrical appK- cations of quaternions is the differentiation of a scalar Amotion of p, the vector of any point in space. Let F{p) = (7, where jP is a scalar function and C an arbitrary constant, be the equation of a series of surfaces. Its differential, /(/),rf/))=0. 96 QUATERNIONS. [CHAP. IV. ia, of course^ a scalar ftinction : and^ being homogeneous and linear in dp, § 130, may be thus written, | Svdp = 0, I where v is a vector, in general a function of p. This vector, p, is easily seen to have the direction of the normal to the given surface at the extremity of p ; being, in ' fact, perpendicular to every, tangent line dp, §§ 36, 98. Its | length, when J' is a surface of the second degree, is the re- I ciprocal of the distance of the tangent-plane from the origin. | And we will show, later, that if p = ix-^jy+iz, I EXAMPLES TO CHAPTER IV. 1. Showtliat (a.) d.8Uq = S.UqV^- = "^ i^ ^'^*- ,dq (e.) d\Tq = {8\dqq-^--8.{dqqr'y}Tq=i -TqV 2. If Fp = :2.8apSpp + lffp^ give dFp = Svdp, show that i; = 2 r.ap)3 + (^ + 2 Safi)p. CHAPTEK V. THE SOLUTION OP EQUATIONS OP THE FIKST DEGREE. 138. ^J^TE have seen that the diflterentiation of any fdnc- ▼ ▼ tion whatever of a quaternion, q, leads to an equation of the form where/^is linear and homogeneous in dq. To complete the pro- cess of differentiation, we must have the means of solving this equation so as to be able to exhibit directly the value of dq. This general question is not of so much practical importance as the particular case in which j^ is a vector ; and, besides, as we proceed to show, the solution of the general question may easily be made to depend upon that of the particular case ; so that we shall commence with the latter. The most general expression for the function/ is easily seen to be ^ ^^(y^ dj) = 2 Kadqb + S.cdq, where a, 6, and e may be any quaternion ftmctions of q what- ever. Every possible term of a linear and homogeneous fimc- tion is reducible to this form, as the reader may easily see by writing down all the forms he can devise. Taking the scalars of both sides, we have Sdr = 8.cdq = SdqSc +8.VdqVc. But we have also, by taking the vector parts, Vdr ^'S.V.adqb = Sdq.'lVab-Y'2.V.a{Vdq)b. 98 QUATERNIONS. [chap. V. Eliminating Sdq between the equations for Sdr and Vchr it is obvious that a linear and vector expression in Vdq will remain. Such an expression, so far as it contains Vdq^ may always be reduced to the form of a sum of terms of the type a5./3 Vdq^ by the help of formulae like those in §§ 90, 91. Solving this, we have Vdq, and Sdq is then found from the preceding equation. 139. The problem may now be stated thus. Find the value of p from the equation aSpp + a^Sfi^p-^... = l.aSpp = y, where o, /3, Oj, i3i, ...y are given vectors. [It will be shown later that the most general form requires but three terms^ i. e. six vector constants a, /3, Oi, /3i, a„ /S, in all.] If we write, with Hamilton, it>p = l.aSpp, the given equation may be written or p = il>-'y, and the object of our investigation is to find the value of the inverse function <^-*. 140. We have seen that any vector whatever may be ex- pressed in terms of any three non-coplanar vectors. Hence, we should expect a priori that a vector such as <t>^p, or <^'p, for instance, should be capable of expression in terms of p, <^p, and <^*/). [This is, of course, on the supposition that p, <^p, and 0*p are not generally coplanar. But it may easily be seen to extend to this case also. For if these vectors be generally coplanar, so are <^p, <J)'p, and <p^p, since they may be written <r, <^(r, and <f>'(r. And thus, of course, <^'p can be expressed as above. If in a particular case, we should have, for some definite vector p, <^p = gp where ^ is a scalar, we shall obviously have <^*p = g^p and 0'p = g^py so that the equation will still subsist. And a SECT. 141.] SOLUTION OP EQUATIONS. 99 similar explanation holds for the particular case when (for some defmite value of p) p, <^p, and ^'p are coplanar. For then we have an eqiiation of the form which gives ^«p = Ail>p + B<p^p = ABp'{-{A-\-B^)4>p. So that </>'p is in the same plane.] If, then, we write — <>V = ^P+y#+'8^*V> (1) it is evident that x^y^z are quantities independent of the vector p, and we can determine them at once by processes such as those in §§ 91, 92. If any three vectors, as i,y, i, be substituted for p, they will in general enable us to assign the values of the three coefficients on the right side of the equation, and the solution is complete. For by putting <^"*p for p and transposing, the equation becomes — a?</)-V = yp + z<l>p'^<l>*p; that is, the unknown inverse function is expressed in terms of direct operations. If x vanish, while y remains finite, we sub- stitute <f>-^p for p, and have -y<t>''p:=zp + <t>p, and if x and y both vanish z<l>-^p = p. 14L To illustrate this process by a simple example we shall take the very important case in which <^ belongs to a central surface of the second order ; suppose an ellipsoid ; in which case it will be shown (in Chap.VIII.) that we may write <^p = — fit* iSip — b^j8;p — c* iSip. Here we have (f}i = aH, <f>H = a*i, <I>H = aH, 4>; = b'j, 4>^j = b'j, 4>'j=^i'jy o 2 100 QUATBBNIONS. [cHAP. V. Hence, putting separately i^j, k for p in the equation (1) of last section^ we have — «• = x+ya* + i»% — <?• s: x+yc*+zc*. Hence a% J% c* are the roots of the cubic which involves the conditions 2:=— («• + *•+<?•), Thus, with the above value of <!>, we have ^»p = a»J»c«p— (a•J» + iV+c»a•)0p^-(a» + i* + (?•)<>V• 142• Putting ^-*<r in place of p (which is any vector what- ever) and changing the order of the terms, we have the desired inversion of the function <f> in the form or the inverse function is expressed in terms of the direct function. For this particular case the solution we have given is complete, and satisfactory ; and it has the advantage of pre- paring the reader to expect a similar form of solution in more complex cases. 143. It may also be useful as a preparation for what follows, if we put the equation of § 140 in the form = {(</>-«')(*-J')(*-^')}p .' (2) This last transformation is permitted because <f> is commutative with scalars like a% i. e. 0(«'p) == «'0p- SECT. 145.] SOLUTION OP EQUATIONS. 101 Here we remark that (by § 140) the equation V.ixl>p = 0, or </)p = ffpf where ff is some undetermined scalar^ is satisfied^ not merely by every vector of null-length, but by the definite system of three rectangular vectors Ai, Bj, Ck whatever be their tensors, the corresponding particular values of ^ being a*, }», c*. 144. We now give Hamilton's admirable investigation. The most general form of a Unaar and vector fimction of a vector may of course be written as ^p = 2 y^qpr, where q and r ai-e any constant quaternions. Hence, operating by S.v where a is any other vector, S<T<l>p = 2S.(rr.qpr=:2S.pr.r(rq^Spit/<r, (3) if we agree to write 4>^ar = iV.ra-q. The functions <f> and (j/ are thus eofyugate to one another, and on this property the whole investigation depends. 145. Let X, /A be any two vectors, such that Operating by 8X and 8,ii we have 5X0P = 0, fiJLu^p = 0. But, introducing the conjugate function 0^, these become 8p4f\ = 0, iSp^V = 0, and give p in the form mp = F</)'A<^V> where ^ is a scalar which, as we shall presently see, is inde- pendent of \, fi, and p. But our original assumption gives P = <>-*FVr hence we have OT<^-' FA/i = F<>'X0V, (4) and the problem of inverting ^ is solved. 102 QUATSBNIONS. [chap. V. (5) 146* It remains to find the value of tlie constant m, and to express the vector as a function of FA/a. Operate on (4) by 8.(f/vf where v is any vector not cophtnar with X and fj,, and we get mS.<l/pil>-^ FV = fnJS.v<t><ir^VKii (by (3) of § 144) = mS.\iiv= S.<l/\it/iMl/v, or "* "" 8.\,xp [That this quantity is independent of the particular vectors \, IX, via evident from the fact that if \'=jpX+j/i+ry, M'=i'A+S'iM+^i»'* and i/=p^k + q^fi+r^v be any other three vectors (which is possible since X, ^i, v are not coplanar)^ we have <t>h^ = p4>k + j^V + r^fv, &c., &c. ; from which we deduce 84'Xitfiij^Vy 8.\ixVf so that the numerator and denominator of the fraction which expresses m are altered in the same ratio. Each of these Quan- tities is in fact an Invariant, and the numerical multiplier is the same for both when we pass from any one set of three vectors to another.] 147. Let us now change <^ to <>+^j where ^ is any scalar. It is evident that <^' becomes <^'4-^, and our equation (4) be* comes >'xVm>v= P i r Pi ?i ri P> ?. r* 5.X'mV= P i r Px ix n P* i> ♦•. SECT, 148.] SOLUTION OP EQUATIONS. 103 = F^'A^V -^g r{4>'\fi + X0V) -e^' FA/a, = (»^<J)-*-|-^X+^'')F^M suppose. In the above equation is what m becomes when </> is changed into </>+^; ^i and ^^ being two new scalar constants whose values are mi = ^ 9 iS'.Afxr Substituting for Ug, and equating the coefficients of the various powers of ^ after operating on both sides by (f>-\-g, we have two identities and the following two equations^ [The first determines x> ai^d shows that we were justified in treating F(<^'A/i+A<^V) ^ ^ linear and vector function of FA/a, The result might have been also obtained thus, SAxVXfx = 5.A^'A/A = -&Afi0'A = -/S.A^FA^, 8.IXxVXtA = 8.lik<t>'fA=z^S.IX(j>V\lM, = maS\fiv—S.\fjxl/v = 5.v(w,FA/A-<>FX/ui); and all three are satisfied by X = «^,-0.] 148. Eliminating x &om these equations we find Ml = (^(^a — <^) + W0~S or «»<^- ' = l»i — «»j<;f) -f <J)% 104 QUATERNIONS. [cHAP. V, which contains the complete solution of linear and vector equa- tions. 149. More to satisfy the student of the validity of the above investigation^ about whose logic he may at first feel some diffi- culties^ than to obtain easy solutions^ we take a few very simple examples to begin with: and we append for comparison easy solutions obtained by methods specially adapted to each case. 150. Example!. Let <^p =: V.apfi = y. Then ^'p = V.fipa = ^p. Hence m = -^^ S{V.aXp F.ofi/S V.avp). Now \y ii, V are any three non-coplanar vectors ; and we may therefore put for them o, P,yiftke latter be non-coplanar. With this proviso "^^^^^^^'^-^'-^-"^^^ m^ = -=-— 8{a^p.aPKy+a.apKr.ayP + a^p.p.V.ayfi) o.apy Hence a^P*SaP4-'yz^^a*P^8afi.p^ ^a*fi^y+SapV.ayP+ r.a{V.ayp)fij | which is one form of solution. By expanding the vectors of products we may easily reduce i it to the form a^p^8afi.p = - a«^ V + a)3«/Say .+ fia^S^y, a-*/SayH-/3-^/My-y or p= —Saf^^- SBCT. 153.] SOLUTION OP EQUATIONS. 105 15L To verify tbis solution^ we have which is the given equation. 152. An easier mode of arriving at the same solution^ in this simple case^ is as follows : — Operating by 8.a and S.fi on the given equation V.apfi = y, we obtain a^Sfip = Say, fi^8ap=:SPY; and therefore aSpp = a-^Say, pSap = P'^Spy. But the given equation may be written aSPp—pSafi-^-pSap = y. Substituting and transposing we get pSafi = o-»Say+/3-*/8)3y-y, which agrees with the result of § 150. 153. K a, /3, y be coplanar, the above mode of solution is applicable^ but the result may be deduced much more simply. For (§101) 8.aPy=0, and the equation then gives 8.afip=^0, so that p is also coplanar with a, fi, y. Hence the equation may be written apP = y, and at once and this^ being a vector^ may be written = a-'Sp-'y'\-p-'8a-'y-y8a-'p-\ 106 QUATERNIONS. [OHAF. Y. This fonniila is equinaleni to that just given^ bat not eqxuil to it term by term. [The student will find it a good exercise to prove direeUy that^ if Oj ^> y are coplanar, we have 154. Example 11. Let <^p = V.afip = y. Suppose a, p, y not to be coplanar^ and employ them 2i&\,ii,v to calculate the coefficients in the equation for ^~'. We have 8.ir4>p = S.fTcfip = B.pV.traP = S.pif/a. Hence tj/p = V.pap = V.fiap. We have now «»i = •^^iS(a.i3o)8.F./3oy+/3a».0.F./3ay+i8o»./3a/3.y) = 2(iSai3)« + a»/3% ^a = ^;^^(M.^.i3ay + a.)3ai3.y+/3o\^,y) = 3fifa)3. Hence a»/3«/So/3.^-^y = a^fi^Safi.p = (2(&/3)« + a«/3«)y-3&^F.o/3y4- V.apV.aPy, wbich^ by expanding the vectors of products^ takes easily the simpler form a^fi'Sap.p = a*^*y-a)3«flfay + 2/3/S(i^fifay-i3a«/Sr^y. 155. To verify this, operate by V.a^ on both sides> and we have SECT. 156.] SOLUTION OF EQUATIONS. 107 :=za*p^{a8Py-fi8ay'^y8afii)-'{2a8ap^pa^)fi*Say + 2afi^S€Lfi8ay-aa*fi*Spy =za^P'Sap.y, or V.afip = y, 156. To solve the same equation without employing the general method^ we may proceed as follows : — y = F.o/3p = p8ap + r. F(a/3)p. Operating by 8. Vafi we have 8.afiy = S.afipSafi. Divide this by fiia/S, and add it to the given equation. We thus obtain Hence p = ^.a-(y+^), a form of solution somewhat simpler than that before obtained. To show that they agree^ however, let us multiply by a*/3'/Sa)3, and we get a^fi*8afi.p = fiay8aP'\'pa8.afiy. In this form we see at once that the right-hand side is a vector, since its scalar is evidently zero (§ 89). Hence we may write a^fi*8afi.p = F.^ayfifa/3— Vafi8.afiy. But by (3) of § 91, -y/8^.o/3Fa^ + aS'./3(Fa^)y + ^fl^.F(a^)ay+ VafiS.a^y = 0. Add this to the right-hand side, and we have a'P^Sap.p z=,y{{8afiyS.apraP)-a{SapSPy-8.fi{rafi)y) + fi{Sap8ay + 8.V{aP)ay), P % 108 QUATBRNION& [chap. V. But {8apyS.apraP = (iSaj8)»-(ra/3)» = a»/3% Sap8pyS.p{rap)y = SapSpySpaSpyk-p^Say = ^^Say Sap8ay^8.F{ap)ay = 8ap8ay+8ap8ay'~a*8py = 2iai0iSoy-.a«i8)3y; and the substitation of these values renders our equation identical with that of § 154. [If a, p,y he coplanar^ the simplified forms of the expression for p lead to the equation 8afi.^^a''y = y— o-»fifoy-f 2^/So-'/3-»iSay— /3-*/8)3y, which, as before, we leave as an exercise to the student.] 157. Sample III. The solution of the equation leads to the vanishing of some of the quantities m. Before, however, treating it by the general method, we shall deduce its solution from that of rr a already given. Our reason for so doing is that we thus have an opportunity of showing the nature of some of the cases in which one or more of m, m^, i», vanish ; and also of introducing an example of the use of vanishing fractions in quaternions. Far simpler solutions will be given in the following sections. The solution of the last- written equation is, § 154, a^$^8ap.p = a^p^y-aP^8ay^Pa^8py+2p8afi8ay. If we now put 0/3 = e + € where e is a scalar, the solution of the first-written equation will evidently be derived fix)m that of the second by making e gradually tend to zero. We have, for this purpose, the following necessary trans- formations : — a'^jS* = alBK.a^ = (^4.e)(^— c) = ^» — e% SECW. 1580 SOLUTION OP EQUATIONS. 109 afi^Say-^pa^Spy = afi.pSay -^ pa.a8py, = {e + €)fi8ay'^{e^€)a8fiy, = ^(/3/Soy + aSpy) -f € Fyc. Hence the solution becomes (e«_€>)^p = (<?«— €«)y— ^(/354iy+oS)8y)-cFy€ + 2tf/3/8oy, = (<?«-€»)y+<?F.yFa)8-€Fy€, = («•— c')y+^Fy€ + y€"— c/Syc, = ^*yH-^Fyc— €/8y€. Dividing by e, and then putting d = 0, we have Now, by the form of the given equation, we see that 8y€ = 0. Hence the limit is indeterminate, and we may put for it x, where x is any scalar. Our solution is, therefore, p = -F^+^€-»; or, as it may be written, since 8y€ = 0, p = €-»(y+«). The verification is obvious — for we have €p szy-k-X. 158. This suggests a very simple mode of solution. For we see that F€p = F€(p-a?€-0 = y, a constant vector whatever x may be. But the vector sign may now be removed as unnecessary, so that we have €{p-X€-^) = y, or p = €-Hy + ^)j if, and only if, p satisfies the equation Vtp^y. 110 QUATERNI0K8, [cHAP. V. 159* To apply the general method, we may take c, y and cy (which is a vector) for A, \Ly v. We find ^> =s Vp€. Hence m =s 0, «ii = - — S.(€,ey.€»y) = -€% »i, = 0. Hence — c*^ + ^» = 0, or *"* = ^*+*"*^- That is, P = ~i^ ^«y+^€> = 6"*y+jr€, as before. Our warrant for putting X€ as the equivalent ottft'-'O is this : — The equation ^.^^^ may be written F.fFco- = = <rc* — c&<r. Hence, unless o- = 0, we have o* || € = X€. 160. Example IV, As a final example let us take the most general form of <^, which, as will be soon proved, may be ex- pressed as follows : — i>p = aSPp-^-a^Sp^p + a^Sp^p = y. Here fp'p = pSap + PiSa^p + P^Sa^p, and, consequently, taking a, a^, a„ which are in this case non- coplanar vectors, for A, ijl, v, we have m = ^ — S,(fi8ajaL+p^Sa^a'{-p^8a^a) {pBaa^^-P^Sa^a^ + ...){fi8ajii^ + . Suxaia^ S.pp^P, 8,aaia2 Saa Saia Sa^a Scuii iSixiai Sa^ai Saa^ /SaiOs Sa^a^ {ASaa + AiSaia+AtSaiO), SECT. 160.] SOLUTION OF EQUATIONS. Ill where A = fi^aiai Sa^a^ — Sa^ax So,^'^^ = — £•. Fctjaa Fiiiaa Ax =)Sa,ai iSaaj — iSoaiiSxaaj = — /S. FajoFaiaa ^a = Saai jSaitta — SaiaiSaa^ = — iSl Faai FaiOa. Hence the value of the determinant is — {SauiS. FaiOa Fajaa + Sa^aS. Foaa Foia, + Sd^aS. Faa, Faia.) = -iS:a(FoiaaS:aa,aa)*{by § 92 (4)} = -.(fif.aa.aa)^ The interpretation of this result in spherical trigonometry is very interesting. By it we see that m =^8.aaxa^S.pfixfi^. Similarly^ »i= ^ SlaipSaax +/3i&iai -^ p^Sa,ax){pS^a, + PxSaxa,+p,Sa,a,) + &c.] o.oaitta = -^ (&appx{SaaiSaxa^''SaxaxSaa^)+ ) O'daiOa ^ (&a/3^i)S.aF.a,Faaai + ) /?.aaiaa = - -jT^^ [S,a( VfiM^ Faai Fata, + Vp,pS. Va^aVa^a^ + Vfixfi.S. Va^a^ Fa,aa) o.aaia3 + S.ax{VpfixS.VaaxVa,a+ ) + S.a,{Vfifi,8.VaaxVaax + )]; or, taking the terms by columns instead of by rows, = - -^ [S. VfiPx {aS. Vaux Va^ai + a^S. Faa^ FaaO + a^S. Faa, Faa^ + + ], 112 QUATERNIONS. [CHAP. V. Again, or, grouping as before, = - -^ — S [/3(rao,flifta, + Va^aSaa^ + ra,a,Soa)+ ...], * ■8[fi{a8.aa,a,) + ] (§92(4)), And the solution is, therefore, -^S.aaia^S.pPiPt.fji-^y = ^8.cuiiai8.pfiiPt'p [It will be excellent practice for the student to work out in detail the blank portions of the above investigation, and also to prove directly that the value of p we have just found satisfies the given equation.] 16L But it is not necessary to go through such a long process to get the solution — ^though it will be advantageous to the student to read it carefully — ^for if we operate on the equa- , tion by 8.aia2, iS^.a^a, and 8,aa^ we get | S.aia^aSpp = S.a^aiy, \ S.a^aaiSpiP = S^^ay, \ S.aaia^SfitP := 8,aaiy. From these, by § 92 (4), we have at once The student will find it a useful exercise to prove that this is equivalent to the solution in § 160. To verify the present solution we have 8.aaia,8.pp^p, {aSfip + a,8p,p + a,8M = aS.pp^pAaia^y + a^S.^^fi^pS^^ay + a^S.^^pp^SM^y = 8.pp,p,{y8.aa,a,), by § 91 (3). SBOT. 163.] SOLUTION OF EQUATIONS. 113 162. It is evident^ from these examples^ that for special cased we can usually find modes of solution of the linear and vector equation which are simpler in application than the general process of § 148. The real value of that process however con- sists partly in its enabling us to express inverse functions of <l>, such as (0+^)""* for instance, in terms of direct operations, a property which will be of great use to us later; partly in its leading us to the fundamental cubic <^' — »^a^* + «»i0 — «a = 0, which is an immediate deduction from the equation of § 148, and whose interpretation is of the utmost importance with reference to the axes of surfaces of the second order, principal axes of inertia, the analysis of strains in a distorted solid, and various similar enquiries. 163. When the function (f> is its own conjugate, that is, when for all values of p and a, the vectors for which form in general a real and definite rectangular system. This, of course, may in particular cases degrade into one definite vector, and any pair of others perpendicular to it j and cases may occur in which the equation is satisfied for every vector. Suppose the roots of %=0 (§ 147) to be real and different, then #1 = giPi ^ ^Pi = ^aPa " where p^, p,, p, are definite vectors. #s = 9zP% - Hence gig^Sp^p^ = S.t^p^i^p^ = S.p.^^p^, or = S.p^(l>^p„ because <^ is its own conjugate. But <p^p, = glp^, <^Vi = glpi> 114 QUATERNIONS. [cHAP. V. and therefore which^ as ^, and ^, are bj hypothesis different, requires Spipi = 0. Similarly Sp^p^ = 0, Sp^i = 0. If two roots be equal, as ^,, ^,, we still have, by the above proof, SpiPi = and Spip^ = 0. But there is nothing farther to determine p, and p„ which are therefore any vectors per- pendicular to pi. If all three roots be equal, every real vector satisfies the equa- tion (<^— ^)p = 0. 164. Next, as to the reality of the three directions in this case. Suppose ^,+*a\/— 1 to be a root, and let p^ + o■,^/— 1 be the corresponding value of p, where ^, and i^ are real numbers, p, and 0-, real vectors, and a/— 1 the old imaginary of algebra. Then *(p, + (r,A/^) = (^,+*,A/3T)(p, + <r,x/^), and this divides itself, as in algebra, into the two equations Operating on these by S.<r,j 5.p, respectively, and subtracting the results, remembering our condition as to the nature of ^ 5<r,<^p, = 5p,^„ we have ^a(<^J+Pi) = 0. But, as (Ts and p, are both real vectors, the sum of their squares cannot vanish. Hence it vanishes, and with it the impossible part of the root. 165. When ^ is self-conjugate, we have shewn that the equation has three real roots, in general different &om one another. SKCT. 167.] SOLUTION OP EQUATIONS. 115 H^iee the cubic in tft may be written and in this form we can easily see the meaning of the cubic. For, let pi, p,, p, be such that Then since any vector may be expressed by the equation pS.pxpipt = pAp^tP+pAptpip-^PiS-piPip (§ 91), we see that when the complex operation, denoted by the left- hand member of the above symbolic equation, is performed on p, the first of the three factors makes the term in pi vanish, the second and third those in p, and p, respectively. In other words, by the successive performance upon a vector of the opera- tions <^— ^1, <^— ^„ <^— ^8> it is deprived successively of its resolved parts in the directions of pi, p,, p, respectively ; and is thus necessarily reduced to zero, since pi, pa, p. are (because we have supposed ffi, ^„ ^, to be distinct) rectangular vectors. 166. If we take pi, p„ p, as rectangular unU-Yecbois, we have — p = piSpip-^-p^Sp^p-^p^Sp^p, whence 4>p = — ^ipj^pip —gtptSp^p SftptSp^p ; or, still more simply, putting i,j\ k for p,, p„ p,, we find that cmy self-conjugate function may4)e thus expressed <lip ^-giiSip—gJSjp-^g^kSkp, provided, of course, i,j, k be taken as roots of the equation Vp4^p = 0. 167. A very important transformation of the self-conjugate linear and vector function is easily derived from this form. We have seen that it involves three scalar constants only, viz- gi} g\9 ^8« I^t ^ enquire, then, whether it can be reduced to the following form # =pP'^qV.{i'\-ek)p{i'-'ek), which also involves but three scalar constants j9, q, e. Q2 116 QUATEBNIOKa [CBXP. V. Sabstituting for p the equiyalent p = ^iSip^jSjp-^kSkpy expanding, and equating coefficients of ij j, h in the two ex- pressions for ^py we find -^i=-/' + ?(2- ! + <?»), -y, =-j?-j(l-<r«). These gire at once —(^2—^3)= ^e*q. Hence, as we suppose the transformation to be real, and therefore «» to be positive, it is evident that gx^g% and ^,—^8 have the same sig^ ; so that we must choose as auxiliary vectors in the last term of ^p those two of the rectangular directions i,y, k for which the coefficients g have the greatest and least values. We have then gx-g^ i = -\{gx-g^y and jD = i(^i+ii's). 168. We may, therefore, always determine definitely the vectors X, /m, and the scalar jd, inr the equation ^p:=PP'\-V,\p\k when ^ is self-conjugate, and the corresponding cubic has not equal roots, subject to the single restriction that is known, but not the separate tensors of X and p.. This result is, important in the theory of surfaces of the second order, and will be developed in Chapter VII. 169. Another important transformation of </> when self-con- jugate is the following, <^p = aa Vap + h^S^py SBCT. 171.] SOLUTIOK OP BQFATIONS. 117 where a and b are scalars^ and a and fi are unit-yectors. This^ of course^ involves six scalar constants^ and belongs to the most general form where pi, p^, ps are the rectangular unit-vectors for which p and <t>p are parallel. We merely mention this form in passing, as it belongs to the /ocal transformation of the equation of surfaces of the second order, which will not be farther alluded to in this work. It will be a good exercise for the student to determine a, j8, a and 8, in terms of ^i, ^„ ^j, and pi, p,, p,. 170, We cannot afford space for a detailed account of the singular properties of these vector functions, and will content ourselves with the enunciation and proof of one or two of the most important. In the equation «»0-«rx|A = r</)'x0V (§ 145), substitute X for ^^X and n for <f/[x, and we h*ave Change <^ to <^+^, and therefore <^' to ^'+y, and m to %, a formula which will be found to be of considerable use. 17L Again, by § 147, Similarly ^8.p{^i,+h)-^p = ^^p^-'p + 5px)»+ V- Hence ^«.p(<^+^)-V - f &p(* + A)-«p = {g-h){p^- ^^} . 118 QUATERKIONS. [CHAF. V. That is^ the fanctions ^M*+^)-V> and ^Ap(<^+*)-ip are identical, i. e. wien equaled to eoMiawU represent the same series of surfaces, not merely when but also, whatever be g and i, if they be scalar functions of p which satisfy the equation This is a generalization, due to Hamiltcm, of a singular result obtained by the author *. 172. The equations ^•p(*+^)-V = o,-i ^jj are equivalent to m8p<l>-*p+gSpxp+g*p* = 0^ mSpip-^p-i-iSpxp-^i^p' = 0. Hence «(1— ar)i8jp<^-*p+(^— AF)iSpXP+W*""**^)/»* = ^> whatever scalar be represented by x. That is, the two equations (1) represent the same surface if this identity be satisfied. As particular cases let (1) 0? = 1, in which case (2) y— ^ar = 0, in which case or mSp^^i^r^p'^gh = 0. * Note on the Cartesian equation of the Wave-Surface. Quarterly Mathem, Journal, Oct. 1859. SECT. 173.] SOLUTION OF BQUATIONS. 119 (3) ^=j^> giving or m{A +ff)8p<f}-^p -k-ghSpyj) = 0. 173. In various investigations we meet with the quaternion where a, fi, y are three unit- vectors at right angles to each other. It admits of being put in a very simple form^ which is occasionally of considerable importance. We have^ obviously, by the properties of a rectangular unit- system J _ py^^ ^ y^^p ^ ^p^^ As we have also &ai3y = -l (§71(13)), a glance at the formulae of § 147 shows that at least if <^ be self-conjugate. If it be not, then (as will be shown in § 174) 4>p = 4/p+ F€p, and the new term disappears in 8q. We have also, by § 90 (2), Fq = a{8p4>y'^SY<lip)+p{SY4>a-'8cufyy)-^y{Sa<liP'-Sfi<tia) = a8.p€y+pS.y€a-\-y8M€p = — (a&ic+i3fil3€ + ySyc) = €. Many similar singular properties of ^ in connection with a rectangular system might easily be given ; for instance, r{aF<l>fi4>y^pr4yy<lia+yF4>a4>fi) = ij>€; which the reader may easily verify by a process similar to that •just given, or (more directly) by the help of § 145 (4). A few 120 QUATERNIONS. [CHAP. V. others will be found among the Examples appended to this Chapter. 174. To conclude^ we may remark that as in many of the immediately preceding investigations we have supposed <^ to be self-conjugate^ a very simple step enables us to pass from this to the non-conjugate form. For, if ^' be conjugate to if>, we have 8p(l/a = 8a'<f>pf and also SpifHr = Satj/p. Adding, we have Sp(4> + 4>>-Sa{4>+<t/)p; 80 that the function (0-1-^0 ^ self-conjugate. Again, Sp4>p = Spff/p, which gives 'Sp{<t>^40p = ^• Hence {<t>^4>')p = ^^Pt where c is some real vector, and therefore Thus every noU'-conjugate linear and vector function differs from a conjugate function solely by a term of the form r^p. The geometric signification of this will be found in the Chapter on Kinematics. 175. We have shown, at some length, how a linear and vector equation containing an unktiown vector is to be solved in the most general case; and this, by § 138, shows how to find an unknown quaternion from any sufficiently general linear equation containing it. That such an equation may be suffi- ciently general it must have both scalar and vector parts : the* SBCT. 178.] SOLUTION OP EQUATIONS. 12l first gives one^ and the second three j scalar equations ; and these are required to determine completely the four scalar elements of the unknown quaternion. 17a Thus Tq^a being* but one scalai^ equation^ gives q=saUr, where r is any quaternion whatever. Similarly Sq = a gives q = a + 0, where is any vector whatever. In each of these eases^ only one scalar condition being given, the solution contains three scalar indeterminates. 177. Again, the reader may easily prove that where a is a given vector, gives Faq=iP+xa. Hence, assuming Saq ^y, We have aq ^y-j-xa+fi, or J = ar+jfa~* + o~*)8. Here, the given equation being equivalent to two scalar con- ditions, the solution contains two scalar indeterminates. 178. Next take the equation Faq^fi. Operating by Ao-*, we get 8q = 8a-' p, so that the given equation becomes F'a(&-^i3+rj) = /3, or . FaFq = p-^a8a''P = af^a'^fi. B 122 QUATERNIONS. [cHAP. V. Prom this, by § 168, we see that ry=a-Har+aF'a-»^), whence q — Sa-^fi-^ a-' {x + aFit-'^) and, the given equation being equivalent to three scalar con- ditions, but one undetermined scalar remains in the value of q. This solution might have been obtained at once, since our equation gives merely the vector of the quaternion aq, and leaves its scalar undetermined. Hence, taking x for the scalar, we have aq = Saq-i- Vaq = x+fi, VI9. Finally, of course, from aq=zp, which is equivalent to four scalar equations, we obtain a definite value of the unknown quaternion in the form q = a-'p. 180. Before taking leave of linear equations, we may mention that Hamilton has shown how to solve any linear equation containing an unknown quaternion, by a process analogous to that which he employed to determine an unknown vector from a linear and vector equation ; and to which a large part of this Chapter has been devoted. Besides the increased complexity, the peculiar feature disclosed by this beautiful discovery is that the symbolic equation for a linear quaternion function, cor- responding to the cubic in c^ of § 162, is a biquadratic, so that the inverse function is given in terms of the first, second, and third powers of the direct function. In an elementary work like the present the discussion of such a question would be out of place : although it is not very difficult to derive the more general result by an application of processes already ex- SECT. 182.] SOLUTION OF EQUATIONS. 123 plained* The reader is therefore referred to the EkmenU of -QuatemionSy p. 491. 181, The solution of the following frequently-occurring par- ticular form of linear quaternion equation aq-^-qh = <?, where a, b, and c are any given quaternions^ has been effected by Hamilton by an ingenious process, which was applied in § 133 (5) to a simple case. Multiply the whole by Ka, and into b, and we have T^a.q^- Ka.qb = Ka,c, and aqb + qb^ = cb. Adding, we have q{T^a + 6* -f 2 8a,b) = Ka.c + cb, from which q is at onde found. To this form any equation such as a'qV'{'c'qcF= e' can of course be reduced, by multiplication by (dY^ and into 182. As another example, let us find the differential of the cube root of a quaternion. If J* = r we have q^dq + qdq,q + dq.cj^ = dr. Multiply by q, and into q-^, simultaneously, and we obtain q^dq-q'^-^q^dq + qdq.q = qdr.q~^. Subtracting this from the preceding equation we have dq»q^—q^dq,q~^ = dr—qdr,q-^, from which dq, or d(r^), can be found by the process of last section. The method here employed can be easily applied to find the differential of any root of a quaternion. R 7, 124 QUATERNIONS. [CHAF. V. 183. To show some of the oharacteristie peouliarities in the solution even of quaternion equations of the first d^;Tee when they are not sufficiently general^ let us take the very simple one and give every step of the solution, as practice in transforma- tions. Apply Hamilton's process (§181), and we get T^a,q = Ka.qb, qb* = aqb. These give q{T'a+b*-^2b8a) = 0, so that the equation gives no real finite value for q unless T*a-\-b^^2bSa = 0, or b^^Sa^pTVa where p is some unit- vector. By a similar process we may evidently show that a=zSb + aTrb, a being another unit- vector. But, by the given equation, Ta = Tb, or 8^a+T^Fa=:8^b + T^Fb; from which, and the above values of a and i, we see that Sa 8b Wa'^Wb^''' '^PP^'"- If, then, we separate q into its scalar and vector parts, thus the given equation becomes (a + a)(r + p) = (/• + p)(a + i9) .'.J. (I) Multiplying out we have r(o-^) = pfl-ap. SECT; 184,] SOLUTION OF EQUATIONS. 125 which gives 8{v^'^fi)p = 0, and therefore p^s Fy{a''P)y where y is an undetermined vector. We have now r{a-p) = p/3-ap = -(a-^)^(a-h^)y. Having thus determined r, we have ?=-^(a+/3)y+ry(a-/3) 2?=-(a+/3)y-y(a-h/3) + y(a-^)-(a-0)y = — 2ay — 2y^. Here, of course, we may change the sign of y, and write the solution of aq^qb in the form q = ay+yp, where y is any vector, and a = UVa, p = ?7ri. To verify this solution, we see by (1) that we require only to show that aq = jj3. But their common value is evidently — y+ay^. It will be excellent practice for the student to represent the terms of this equation by versor-arca, as in § 54, and to deduce the above solution from the diagram directly. He will find that the solution may thus be obtained almost intuitively. 184. No general method of solving quaternion equations of the second or higher degrees has yet been found; in fact, as will be shown immediately, even those of the second degree involve (in their most general form) algebraic equations of the 126 QUATERNIONS. [cHAP. V. sixteenth Aegre^. Hence, in the kvf remaining section? of tim Chapter we shall confine ourselves to one or two of ilie' simple forms for the treatment of which a definite process has been devised. But first, let us consider how many roots an equation of the second degree in an unknown quaternion must generally have. If we substitute for the quaternion the expression w^-ix-^jy+kz (§ 80), and treat the quaternion constants in the same way, we shall have (§ 80) four equations, generally of the second degree^ to determine w, x, y, z. The number of roots will therefore be 2* or 16. And similar reasoning shows us that a quaternion equation of the «»th degree has m* roots. It is easy to see, however, from some of the simple examples given above (§§ 175 ^178, &c.) that, unless the given equation is equivalent to four scalar equations, the roots will contain one or more indeterminate quantities. 185. Hamilton has effected in a simple way the solution of the quadratic j« = qa-^rby or the following, which is virtually the same (as we see by taking the conjugate of each side). He puts S' = i(^+^+p) where «e; is a scalar, and p a vector. Substituting this value, we get a^ -{■{w-^pY-^'^wa + ap-^'pa = 2{a^-\-wa'\-pa)-\-^b, or {w-{'py-{'ap'-pa = a'-h4i. If we put Va = a, %»-f 4i) = c, V{a^ + ^b) = 2y, this be- ^^"'^^ («;+/>)'-h2rap = ^+2y; SECST. la?.] SOLUTION OF EQUATIONS. 127 wfaich^ by equating separately the scalar and vector parts^ may be broken up into the two equations The latter of these can be solved for p by the process of § 156, or more simply by operating at once by 8,a which gives the value of 8{W'^a)p. If we substitute the resulting value of p in the former we obtain, as the reader may easily prove, the equation The solution of this scalar cubic gives six values of w, for each of which we find a value of p, and thence a value of q, Hamilton shows {Lectures , p. 633) that only two of these values are real quaternions, the remaining four being biqua- temions, and the other ten roots of the given equation being infinite. Hamilton farther remarks that the above process leads, as the reader may easily see, to the solution of the two simul- taneous equations and he connects it also with the evaluation of certain continued fractions with quaternion constituents. 186. The equation g^ = aq-k-qby though apparently of the second degree, is easily reduced to the first degree by multiplying by, and into, qr^y when it be- comes l=^ia + J^-i, and may be treated by the process of § 181, 187. The equation 'LA^q'^^aqb, 128 QUATBRNION& [CAAF. V. where a and b are given quatemions and A^ a scalarj gives ^^7 qaqb^aqbq (1) This may be written Iq^aqb) = {aqb)q; &>^d> ^7 § ^^9 it is evident that the planes of q and aqb must coincide. A little fart^ier consideration will show that in general we must have the planes of a, by and q coincident. The solution of such equations th|ks becomes very easy, for the commutative law of multiplication is not violated (§ 54). EXAMPLES TO CHAFFER V. ^ !. Solve the following equations : — {a.) r.app = r.ayp. , («.) apfip = papp. (<?.) ap+pP=: y. {d.) S.aPp+fiSap'-firfip = y. {€.) p + apfi^afi. Does the lasi of these impose any restriction on the generality of a and /3? / 2. Suppose p^i^+jy-k-kzy and 4>p = aiSip + bjSjp -f ckSkp ; put into Cartesian coordinates the following equations :-— (a.) T<t>p = 1. (i.) 8p<l>^p = ^l. {d.) Tp = T.<f>Up. CHAP. VJ EXAMPLES TO CHAPTBB V. 129 3. If kyfjL, V he any three non-coplanar vectors, and show that q is necessarily divisible by SX^uf. Also show that the quotient is where Fep is the non-commutative part of ^p. Hamilton, ElemenU, p. 442. 4. Solve the simultaneous equations : — ^ ^ 4Sp<^p = 0. J Sap = 0,' l'.aipicp= ^«p =0,-. ^ ' 8.aipKp- O.i 5. If # = ^fi8ap+ Vrp, where r is a given quaternion, show that and Lectures, p. 561, 6. If jjoj'] denote pq'-qp, (pq) ... S.plqr-], Ipqr] ... (i>sr)+[rj]^ + MSj+M^, and (pg^^) "' 8. piers'], show that the following relations exist among any five qua- ternions = JO (qrst) + q {rsfp) + r{8tpq) + s{tpqr) + t{pqf8), and jCiw^O = [^*CI^2"" i^iplSrq + [^]%— [jor«]%. Elements, p. 492. / 130 QUATBBNIONS. [cHAF. T. 7. Show that if ^^ ^^ be any linear and vector fdnctions^ and a, P, y rectangular unit- vectors, the vector is an invariant. If ^p = 2i,^p, and # = S«?ifiiriP, show that this invariant may be expressed as -sr#f or srt^iKi. Show abo that ^^p— ^^/> = VOp. 8. Show that if ^p = a8ap'\-pSPp-\-ySypy where a,P,y sire any three vectors, then ^8Kcfiy4-'p = a.Sa.p^p^Sp^p'^y.Sy.py where Oi = F/Sy, &c. 9. Show that any self-conjugate linear and vector function may in general be expressed in terms of two given ones, the expression involving terms of the second order. Show also that we may wri£e where a, b, c, x, y, z are scalars, and «r and cu the two given functions. 10. Solve the equations : — [a.) J* = Sji+ioy. {L) j« = 2^ + i. - {c) qaq ^bq+c. (d,) aq z:z qr = rb. CHAPTER VI. GEOMETRY OP THE STRAIGHT LINE AND PLANE. 188. TTAVING, in the five preceding Chapters, given a •*--■• brief exposition of the theory and properties of quaternions, we intend to devote the rest of the work to ex- «amples of their practical application, commencing, of coarse, with the simplest curve and surface, the straight line and the plane. In this and the remaining Chapters of the work a few of the earlier examples will be wrought out in their fiillest detail, with a reference to the first five whenever a transforma- tion occurs ; but, as each Chapter proceeds, superfluous steps will be gradually omitted, until in the- later examples the fiill value of the quaternion processes is exhibited. 189. Before proceeding to the proper business of the Chapter we make a digression in order to give a few instances of ap- plications to ordinary plane geometry. These the student may midtiply indefinitely with great ease. (a,) Emlid, I. 5. Let a and ^ be the vector sides of an isosceles triangle; j9— a is the base, and Ta = T/3. The proposition will evidently be proved if we show that a{a^P)-^ = Kpifi^a)'^ (§ 52). S 2 132 QUATERNIONS. [CHAP. VI. This gives a(a-^)-* = 03-a)-»ft or (/3-.a)a = i9(a-i9), or — o*=— /3*. (b.) Huelid, I. 32. Let ABC he the triangle, and let AB ^ where y is a unit-vector perpendicular to the plane of the triangle. If I = \, the angle CAB is a right angle (§ 74). Hence ^ = /-(§ 74). Let-B = «»^, C'rr:*-. We have 2 A 2 [7CB = y«[7CZ, UBA = y~?^5C: Hence UBA^ y^.y\'/UAB, or — 1 = y'+«+*. That is ;+«»+^= 2, or A-^B+C=Tr. This is, properly speakings Legendre's proof; and might have been given in a far shorter form than that above. In fact we have for any three vectors, U.^ll^l (§60), which contains Euclid's proposition as a mere particular case. {c.) Euclid, I. 35. Let )3 be the common vector-base of the parallelograms, a the conterminous vector-side of any one of them. For any other the vector-side is a+;r)3 (§ 28)^ and the proposition appears as Trp{a-\-xfi) = TFpa (§§ 96, 98), which is obviously true. SECT. 189.] GEOMETRY OF STRAIGHT LINE AND PLANE. 133 (d.) In the base of a triangle find the point from which lines^ drawn parallel to the sides and limited by them^ are equal. If a, /3 be the sides^ any point in the base has the vector p = (1— ^)o+iP^. Per the required point which determines x. Hence the point lies on the line which bisects the vertical angle of the triangle. This is not the only solution, for we should have written T(l-a?)Ta= TxTfiy instead of the less general form above wAicA tacitly assumes that 1 —0? and X are positive. We leave this to the student. {e.) Jf perpendiculars be erected outwards at the middle points of the sides of a triangle, each being propor- tional to the corresponding side, the mean point of the triangle formed by their extremities coincides with that of the original triangle. Find the ratio of each perpendicular to half the corresponding side of the old triangle that the new triangle may be equi- lateral. Let 2 a, 2)3, and 2(a-hj3) be the vector-sides of the triangle, i a unit- vector perpendicular to its plane, e the ratio in question. The vectors of the corners of the new triangle are (taking the comer opposite to 2)3 as origin) pj = a+eia, p, -2a + p + eip, p, = a-hi3-^i(a+)3). 134 QUATKBKIONS. [cHA^. VI. From these i(Pi +p.4-/>,) = i(4a + 2i3) = i(2a + 2(a + j3)), which proves the first part of the prppositioD. For the second part, we must have Substituting, expanding, and erasing terms common to all, the student will easily find 3^2 ^ j. Hence, if equilateral triangles be described on the sides of any triangle, their mean points form an equilateral triangle. 190. Such applications of quaternions as those just made are of course legitimate, but they are not always profitable. In fact, when applied to plane problems, quaternions often de- generate into mere scalars, and become (§ 33) Cartesian co- ordinates of some kind, so that nothing is gained (though nothing is lost) by their use. Before leaving this class cf questions we take, as an additional example, the investigation of some properties of the ellipse. 19L We have already seen (§31 {i)) that the equation p = acos0+j9sin^ represents an ellipse, $ being a scalar which may have any value. Hence, for the vector-tangent at the extremity of p we have dp ..,«>, r. =— osmd + i3cos^, which is easily seen to be the value of p when is increased by IT -. Thus it appears that any two values of p, for which differs by ~, are conjugate diameters. The area of the parallel- ogram circumscribed to the ellipse and touching it at the extremities of these diameters is, therefore, by § 96, 4yrp^ = 4Tr(acos^-hi3sin^)(-a8ind + ^cos^) a constant, as is well known. SCOT. 194.] GEOMETBT OF 8TBAI0HT LIKE AND PLANE. 135 192. For equal conjugate diameters we must have T(acos^+/38iii^ = y(— asm(?+j3cos(?), or (a«— ^«)(cos*^— 8m*fl) + 4iSaj3costfsind = 0, or tan2g=— ^„ ^ . The square of the common length of these diameters is of course 2~' because we see at once &om § 191 that the simi of the squares of conjugate diameters is constant. 193. The maximum or minimum of p is thus found ; dTp_ I dp = — =-(-(a«-j3«)costfsin^4-'Saj3(cos»^-sin*fl)). For a maximum or minimum this must vanish^ hence 2SaP tan2^ = i«— fl» fi' and therefore the longest and shortest diameters are equally inclined to each of the equal conjugate diameters. Hence^ also^ they are at right angles to each other. dTo [The student must carefully notice that here we put -j^ = 0, up and not ^ = 0. A little reflection will show him that the * latter equation involves an absurdity.] 194. Suppose for a moment a and )3 to be the greatest and least semidiameters. Then the equations of any two tangent- lines are p = acosB -{-fi&mO -f a?(— asind +fico8 0), p = acosdi-f)38in(?i+;ri(— osin(?i-h/8cos(?i). 136 QUATBBNIOlSrS. [cHAP. YL If these tangent lines be at right angles to each other 5(— a8ind+)8co8^)(— asin^i+jQcostfJ = 0, or a* sin sin 01 +j9' cos 0008^1 =: 0. Also, for their point of intersection we have, by comparing coefficients of a, ^ in the above values of p, <x>s0— ;psin0 = cos^i— ^Pisin^i, sin0+;pcosd = BmOi-j-SiCosSi. Determining x^ from these equations, we easily find the equation of a circle ; if we take account of the above relation between and 0^, Also, as the equations above give w = — ;Pi, the tangents are equal multiples of the diameters parallel to them ; £k> that the line joining the points of contact is parallel to that joining the extremities of these diameters. 195. Finally, when the tangents p = acos0 +j9sind +x (— asin^ -^ficoB0), p = acos0i+i9sindi+iri(— asin^i+jScos^i), meet in a given point p = aa+bp, we have a = cos 0^x sin = cos^i — a?i sin 0^, b = Bm0-\'XQOB0 = sin^i+a^iCos^i, Hence a?* = a* -f 5' — 1 = a?; and acostf + isin^ = 1 = dfcos^i+isin^j determine the values of and x for the directions and lengths of the two tangents. The equation of the chord of contact is p = y (a cos 4- )3 sin 0) 4- ( 1 —y) (a cos ^^ + j9 sin 0^). If this pass through the point p =zpa+qp, SECT. 197.] GBOMBTBT OF STRAIGHT LINE AND PLANE. 187 we have p ^^ yooB$+{l — y) cos O^, from which, by the equations which determine $ and 0^, we get Thus if either a and b, or p and q, be given, a linear relation connects the others* This, by § 30, gives all the ordinary properties of poles and polars. 196. Although, in §§ 28-30, we have already given some of the equations of the line and plane, these were adduced merely for their applications to anharmonic coordinates and transversals; and not for investigations of a higher order. Now that we are prepared to determine the lengths and inclinations of lines we may investigate these and other similar forms anew. 197. The equation of the indefinite line drawn through the origin 0, of which the vector OA, = a, forms a part, is evidently or p II a, or Vap = 0, or Up^ Ua; the essential characteristic of these equations being that they are linear, and involve one indeterminate scalar in the value of p. We may put this perhaps more clearly if we take any two vectors, j3, y, which, along with a, form a non-coplanar system. Operating with S.Fafi and 8,Fay upon any of the preceding equations, we get ":::) '■' 8.app = 0, ' and S,ayp Separately, these are the equations of the planes containing a, /3, and a^ y; together, of course, they denote the line of intersection. T 138 QUATBRNlONa [CHAP. VL 198. Conversely, to solve equations (1), or to find p in terms of known quantities, we see that they may be written 8.pFap = 0,^ S.pray = 0,J so that p is perpendicular to Fafi and Fayy and is therefore parallel to the vector of their product. That is, pWFFapFay, II -a8.apy, or p = xa, 199. By putting p— J9 for p we change the origin to a point B where OB = — /3, or BO = )3 ; so that the equation of a line parallel to o, and passing through the extremity of a vector fi drawn from the origin, is or p = j3-f iPcu Of course any two parallel lines may be represented as p = p +a?a, p = fii+af^a; or Fa(j)-^p) = 0, Fa(p^P,) = 0. 200. The equation of a line, drawn through the extremity of j9, and meeting a perpendicularly, is thus found. Suppose it to be parallel to y, its equation is p = P^xy. To determine y we know, Jirst, that it is perpendicular to o, which gives Say = 0. Secondly, a, ^, and y are in one plane, which gives S.apy = 0. These two equations give ylir.ara^. SBOT. 202.] GEOMETRY OP STBAIGHT LINE AND PLANE. 139^ whence we have p = p^xaFap. This might have been obtained in many other wajns; for instance^ we see at once that /3 = a-»a/8 = a-^SaP-^ar^Fap. This shows that ar^Vafi (which is evidently perpendicular to a) is coplanar with a and jS^ and is therefore the direction of the required line ; so that its equation is the same as before if we put — ^-r for x. 20L By means of the last investigation we see that is the vector perpendicular drawn from the extremity of j3 to the line p = xa. Changing the origin^ we see that la-»ra03-y) is the vector perpendicular from the extremity of j3 upon the line p = y+xa. 202. The vector joining £ (where OB = j3) with any point in p = y-^-xa is y+ira— )3. Its length is least when rfr(y+a?a-i3) = 0, or /Sa(y + iPa— /3) = 0, i. e. when it is perpendicul$ur to a. The last equation gives xa^ + Saiy—p) = ^> or ara=— a-*/Sa(y— )3). T 2, 140 QUATBRNIOKS. [ohaf. VI. Hence the vector peipendiculttr is y-^-.o-»Sa(y-^), or QT^Vaiy-p)^ ^ar^Vaffi-y), which agrees with the result of last section. SOS. To find the ahorteerfi vector distance between two Unids and pi=/3i+^iai; we must put dT{p^pi) = 0, or Sip^p^Xdp-dp^) = 0, or S{p^pi){adx+aidxi) = 0. Since or and Xi are independent, this breaks up into the two conditions Sa (p^p,) = 0, Saiip—pi) = ', proving the well-known truth that the required line is perpen- dicular to each of the given lines. Hence it is parallel to Faai, and therefore we have p— Pi 5= /3 + ii?«— )9i— iPjaji ^yVoai* .•. {\) Operate by &aoi and we get This determines y, and the shortest distance required is [Note. In the two last expressions T before 8 is simply inserted to ensure that the length be positive. If &aai(j3— )3i) be negative, then (§ 89) &aia(i3— )3i) is positive. If we omit the T we must use in the text that one of tjiese two expressions which is positive.] To find the extremities of this shortest distance, we must 8BCT. 206.] fVHHMT Of WmtBBttIT LINS AND PLANE. 141 operate on (I) wi<^ JS.a and S^i. We thus obtain two'equations^ wiiidi determine a and^lPo as y is already known. A somewhat different mode of treating this problem will be disciuwed presei^j. 204. The equation 5ap = imposes on p the sole condition of being perpendicular to a; and therefore^ l>eing satisfied by the veclor drawn from the origin to any point in a plane through the origin and perpen* dicular to a, is the equation of that plane. To find this equation by a direct process similar to that usually employed in coordinate geometry^ we may remark that^ by § 29, we may write where y3 and y are any two vectors perpendicular to a. In this form the equation contains two indeterminates, and is often useful ; but it is more usual to eliminate them, which may be done at once by operating by i&a, when we obtain the equation first written. It may also be written, by eliminating one of the ipdet^er- minates only, as tt^ where the form of the equaUoi^ shows that Skip = 0, 205. Similarly we see that Saip^fi) :== represents a plane drawn through the extremity of fi and per- pendicular to <i. This, of couisse, may, like the last, be put into various equivalent forms. 206. The line of intersection of the two planes S.a(p-p) =0,- and S.a. '(^-^) =•'''1 (1) 142 QUATERNIONS. [chap. VL contains all points whose valae of p satisfies both conditions. But we may write (§ 92)^ since a^ a,^ and Faai are not co- planar, pS.aaiVaai = Faai8,aaip+F.aiFaaiSap-j'F,F(€Mi)€LSaip, or, bj the given equations, -py« Toa, = F.a^ Faa^Safi + F. F{(M^)aSa^fi^ + x Faa^, (2) where x, a scalar indeterminate, is put for S.cMip which may have anj value. In practice, however, the two definite given scalar equations are generally more useM than the partially indeterminate vector-form which we have derived from them. When both planes pass through the origin we have )3 = )3i = 0^ and obtain at once _ xFaa as the equation of the line of intersection. 207. The plane passing through the origin^ and through the line of intersection of the two planes (1)^ is easily seen to have the equation Sa,Map^SafiSa,p = 0, or S{aSaiP^''aiSap)p = 0. For this is evidently the equation of a plane passing throngh the origin. And, if p be such that Sap = Sap, we also have Sa^p = Sa^Pi, which are equations (1). Hence we see that the vector aSaipi'-aiSaP is perpendicular to the vector-line of intersection (2) of the two planes (1), and to every vector joining the origin with a point in that line. The student may verify these statements as an exercise. . 208. To find the vector-perpendicular from the extremity of p on the plane ^ _ q SECT. 210.] GEOMETRY OF STRAIGHT LINE AND PLANE. 143 vre must note tliat it is necessarily parallel to a, and hence that the corresponding value of p is where xa is the vector perpendicular in question. Hence &i(j3+a?o) = 0, which gives xa^ = — /Saj3, or xa =— a^*/Sa)3. Similarly the vector-perpendicular from the extremity of p on the plane Sa{p-y) = may easily be shown to be 209. The equation of the plane which passes through the extremities of a, /3, y may be thus found. If p be the vector of any point in it, p— a, a— /3, and /3— y lie in the plane, and therefore (§ 101) 5.(p-o)(a-i3)(^-y) = 0, or Sp( FaP + Ffiy + Fya) - S.apy = 0. Hence, if 8 = x{Fafi+ Fpy+ Fya) be the vector-perpendicular from the origin on the plane con- taining the extremities of a, ft y, we have 8 = (roj3+ Ffiy-^' Fya)-'8.aPy. From this formula, whose interpretation is easy, many curious properties of a tetrahedron may be deduced by the reader. 210. Taking any two lines whose equations are p = ^ -ha?a, p = ft + iz^iai, we see that S,aai{p-^b) = is the equation of a plane parallel to both. JFkick plane, of course, depends on the value of 5. 144 QUAlTEBQillCmS. [OHIp. YI. Kow H istfi, the fhine contains the first fine; if b^Pt, the second. Hence^ if yFaai be the shortest vector distance between the lines^ we have S.aaAP-fii-yFaa,) = 0, or T{yFaa^) ^T8.ifi^p,)Uraa^, the result of § 203. 21L Find the equation of the plane> passing through the origin^ which makes equal angles with three given lines. Also find the angles in question. Let a^ ^^ y be unit- vectors in the directions of the lines, tfnd let the equation of the plane be 8bp 3= 0. Then we have evidently 8aJb s Spi s 8yh = x, suppose^ where x is the sine of each of the required anglei^. But (§ 92) we have h8.aPy = x{rafi+ Vpy^ Vya). Hence 8.p{ Fap -f Fpy + Fya) = is the required equation ; and the required sine is T(fafi+Fpy+Fyay 212. Find the locus of the middle i)oint6 of a series of straight lines^ each parallel to a given plane and having its extremities in two fixed lines. Let 8yp s be the plane, and the fixed lines. Also let x and a?i correspond to the extremities SECT. 214.] GEOMETKY OF STRAIGHT LINE AND PLANE. 145 of one of the variable lines, cr being the vector of its middle point. Then, obviously, Also Syifi—Pi-j-wa—Xia^) = 0. This gives a linear relation between x and a?i, so that, if we substitute for x^ in the preceding equation, we obtain a result of the form • » • ,«, where h and € are known vectors. The required locus is, there- fore, a straight line. 213, Three planes meet in a point, and through the line of intersection of each pair a plane is drawn perpendicular to the third; prove that, in general, these planes pass through the same line. Let the point be taken as origin, and let the equations of the planes be Sap = 0, 8l3p = 0, 8yp = 0. The line of intersection of the first two is || Vafi, and therefore the normal to the first of the new planes is F.yFap. Hence the equation of this plane is S^pKyFap = 0, or SfipSay-'SapSfiy = 0, and those of the other two planes may be easily formed from this by cyclical permutation of a, /3, y. We see at once that any two of these equations give the third by addition or subtraction, which is the proof of the theorem. 214. Given any number of points A, J5, C, &c., whose vectors (from the origin) are a^, aj, Oj, &c., find the plane through the origin for which the sum of the squares of the perpendiculars let fall upon it from these points is a maximum or minimum. u 146 QUATERNIONS. [CHAP. VI. Let Svp = be the required equation, with the condition (evidently allowable) ?W = 1. The perpendiculars are (§ 208) — flr-*iS«rai, &c. Hence 2 S* fsra is a maximum. This^ves 1.8rffa8adisr = ; and the condition that tr is a unit- vector gives Siffdw ^ 0. Hence, as thr may have any of an infinite number of values, these equations cannot be consistent unless where a? is a scalar. The values of a are known, so that if we put ^.aSavr = (fyar, <^ is a given self-conjugate linear and vector function, and there- fore X has three values {g^g^y g^y § 164) which correspond to three mutually perpendicular values of -or. For one of these there is a maximum, for another a minimum, for the third a maximum-minimum, in the most general case when^i,^a,^s are all different. 215. The following beautifiil problem is due to Maccullagh. Of a system of three rectangular vectors, passing through the origin, two lie on given planes, find the locus of the third. Let the rectangular vectors be «r, p, o-. Then by the con- ditions of the problem 5Wp = 8p(r = ScT'Bj = 0, and Swta = 0, S^p = 0. The solution depends on the elimination of p and ts among these five equations. [This would, in general, be impossible, as p SECT. 215.] GEOMETRY OF STRAIGHT LINE AND PLANE. 147 and -BT between them involve sijc unknown scalars ; but, as the tensors are (by the very form of the equations) not involved, the five given equations serve to eliminate the four unknown scalars which are really involved. Formally to complete the requisite number of equations we might write Tur = a, Tp=: b, but a and b may have any values whatever.] Prom Sa'sr = 0, fir-cr = 0, we have -bt = xFaa. Similarly, from Spp = 0, Sap = 0, we have p = yV^a, Substitute in the remaining equation Srffp = 0, and we have S.FaaFptr = 0, or SatrSpa-'-a^Safi = 0, the required equation. As will be seen in next Chapter, this is a cone of the second order whose circular sections are perpen- dicular to a and /3. [The disappearance of x and y in the elimination instructively illustrates the note above.] u 2, 148 QUATERNIONS. [cHAP. VI. EXAMPLES TO CHAPTER VI. 1. What propositions of Euclid are proved by the m&rQform of the equation p = (1— a?)o+a?j3, which denotes the line joining any two points in space ? 2. Show that the chord of contact, of tangents to a parabola which meet at right angles^ passes through a fixed point. 3. Prove the chief properties of the circle (as in Euclid , III) from the equation p = acos^ + i^sind; where Ta = 1)3, and Safi = 0. 4. What locus is represented by the equation where Ta=l? 5. What is the condition that the lines Top = ft Fa,p = fi„ intersect ? If this is not satisfied, what is the shortest distance between them ? 6. Knd the equation of the plane which contains the two parallel lines Faip-^P) = 0, ra(p-ft) = 0. 7. Find the equation of the plane which contains Faip^P) = 0, and is perpendicular to Syp = 0. 8. Find the equation of a straight line passing through a given point, and making a given angle with a given plane. Hence form the general equation of a right cone. CHAP. VI.] EXAMPLES TO CHAP, VI. 149 9. What conditions must be satisfied with regard to a number ^of given lines in space that it may be possible to draw through a • ', each of them a plane in such a way that these planes may ' ^^ f intersect in a common line ? '. 10. Find the equation of the locus of a point the sum of the squares of whose distances from a number of given planes is constant. 11. Substitute 'Mines'' for ''planes'' in (10). i/42. Find the equation of the plane which bisects, at right angles, the shortest distance between two given lines. Find the locus of a point in this plane which is equidistant fK)m the given lines. .13. Find the conditions that the simultaneous equations Sap = a, Sfip = J, Syp = (?, may represent a line, and not a point. y 14. What is represented by the equations iSapY = [S^pY = (%)% where a, )3, y are any three vectors ? ^15. Find the equation of the plane which passes through two given points and makes a given angle with a given plane. /l6. Find the area of the triangle whose comers have the vectors a, )3, y. Hence form the equation of a circular cylinder whose axis and radius are given. ^7. (Hamilton, Bishop Law^ 8 Premium Ex., 1858). V {a,) Assign some of the transformations of the expression Tap where a and ^ are the vectors of two given points A and B. * yS.) The expression represents the vector y, or 0(7, of a point C in the straight line AB. {c.) Assign the positioin of this point C, ^^ *v.^ :r. 5^^v ^ 150 QUATERNIONS. [chap. VI. 18. (Ibid.) (a.) If a, Pjy,h he the vectors of four points A, B, C, D, what is the condition for those points being in one plane? (J.) When these four vectors from one origin do not thus terminate upon one plane^ what is the expression for the volume of the pyramid, of which the four points are the corners ? {c.) Express the perpendicular b let fall from the origin on the plane ABC, in terms of a, fi, y. A9. Find the locus of a point equidistant from the three planes Sap = 0, Spp = 0, Syp = 0. 20. If three mutually perpendicular vectors be drawn from a point to a plane, the sum of the reciprocals of the squares of their lengths is independent of their directions. 21. Find the general form of the equation of a plane from the condition (which is to be assumed as a definition) that any two planes intersect in a single straight line. l/2^ CHAPTEK VII. THE SPHERE AND CYCLIC CONE. 4 FTER that of the plane the equations next in order 216. -L^. Qf simplicity are those of the sphere, and of the cone of the second order. To these we devote a short Chapter as a valuable preparation for the study of surfaces of the second order in general. 217. The equation Tp = Ta, or p^ = a% denotes that the length of p is the same as that of a given vector a, and therefore belongs to a sphere of radius Ta whose centre is the origin. In § 107 several transformations of this equation were obtained, some of which we will repeat here with their interpretations. Thus /SCp-f a)(p-a) = shows that the chords drawn from any point on the sphere to the extremities of a diameter (whose vectors are a and —a) are at right angles to each other. y(p-ha)(p-a) = 2TFap shows that the rectangle under these chords is four times the area of the triangle two of whose sides are a and p. p = (p-|-a)-*a(pH-a) (see § 105) shows that the angle at the centre in any circle is double that 152 QUATERNIONS. [chap. VIL at the circumference standing on the same arc. All these are easy consequences of the processes already explained for the interpretation of quaternion expressions. 218. If the centre of the sphere be at the extremity of a the equation may be written T{p^a) = Tfi, which is the most general form. If Ta = Tpy or a« = ^% in which case the origin is a point on the surface of the sphere^ this becomes p» - 2 iSap = 0. From this, in the form 8p{p-2a) = another proof that the angle in a semicircle is a right angle is derived at once. 219. The converse problem is — Find the locus of the feet of perpendiculars let fall from a given point (p = /3) on planes passing through the origin. Let Sap = be one of the planes, then (§ 208) the vector-perpendicular is —a-' Sap, and, for the locus of its foot, p = p-^a-^Sap =:a-'rafi. [This is an example of a peculiar form in which quaternions sometimes give us the equation of a surface. The equation is a vector one, or equivalent to three scalar equations; but it involves the undetermined vector a in such a way as to be equivalent to only two indeterminates (as the tensor of a is SECT. 221.] THE SPHERE AKD CYCLrc CONE. 153 evidently not involved). To put the equation in a more imme-* diately interpretable form, a must be eliminated, and the re- marks just made show this to be possible.] Now {p-Py = a-'S^afi, and (operating by 8.p) Adding these equations^ we get p'Sfip = 0, or ^\ _ mP np-i)=^f' so that, as is evident, the locus is the sphere of which )3 is a diameter. 220. To find the intersection of the two spheres T(p-a) = Tp, and r(p-aO= %, square the equations, and subtract, and we have which is the equation of a plane, perpendicular to o— aj the vector joining the centres of the spheres. This is always a real plane whether the spheres intersect or not. It is, in fact, what is called their Radical Plane. 221. Find the locus of a point the ratio of whose distances from two given points is constant. Let the given points be and A, the extremities of the vector a. Also let P be the required point in any of its positions, and OP z= p. Then, at once, if « be the ratio of the lengths of the two lines, T{p'^a)=^nTp. This gives p*— 2iSap-ha» = «VS X 164 QUATBRNIONS, [cftAF.YIL or> by an easy traiiafonnation^ Thus the locus is a sphere whose radius is T(- ^), and whose centre is at B, where OB = , a definite point in the 1— «» line OA. 222. If in any line^ OP, drawn from the origin to a given plane, OQ be taken such that OQ.OP is constant, find the locus o{Q. Let Sap = a be the equation of the plane, 17 a vector of the required surface. Then, by the conditions, TerTp = constant = i» (suppose), and V^TSF = Dp. Irom these p = -== — = . Tiff -or* Substituting in the equation of the plane, we have aw' + i*iSo«r = 0, which shows that the locus is a sphere, the origin being situated on it at the point farthest from the given plane. 223* Find the locus of points the sum of the squares of whose distances from a set of given points is a constant quantity. Find also the least value of this constant, and the correspond* ing locus. Let the vectors from the origin to the given points be Oj, Oj, a^ and to the sought point p, then -c» = {p^a,y-^{p-^a,y + +{p-CLny, =:»p«-25p2a-f-S(o»). Otherwise . 2a.^_ c'-h2(a') . (2a)' yp-T) u — + "^' SEOT. 22&] THE SPHEBB AKD CTOLIC CONE. 166 the equation of a sphere the vector of whose centre is — , i.e. whose centre is the mean of the system of given points. Suppose the origin to l>e placed at the mean point, the equa- tion becomes ^i _|. 2(a») ^'= ^ — ' The right-hand side is negative^ and therefore the equation denotes a iipal surface^ if as might have been expected. When these quantities are equal, the locus becomes a point, viz. the origin. 224. If we differentiate the equation Tp=z To. we get Spdp = 0. Hence (§ 137), p is normal to the surface at its extremity, a well- known property of the sphere. If tsr be any point in the plane which touches the sphere at the extremity of p, cr— p is a line in the tangent plane, and therefore perpendicular to p. So that 8p{w--p) = 0, or i8Wp=— 2>» = a» is the equation of the tangent plane. 225. If this plane pass through a given point j5, whose vector is )8, we have 8pp = o». This is the equation of a plane, perpendicular to )8, and cutting from it a portion whose length is If this plane pass through a fixed point whose vector is y we must have jgpy -- a«, X2» 156 QUATBBNIONS. [chap. VII. 80 that the locos of /3 is a pkne. These results contain all the ordinary properties of poles and polars with regard to a sphere. 226. A line drawn parallel to y^ from the extremity of fi, has the equation p -. p^xy. This meets the sphere p» = o« in points for which x has the values given by the equation p* + ix8py+x^y^ = aK The values of x are imaginary^ that is^ there is no intersection^ if o«y"-H^"/3y < 0. The values are equals or the line touches the sphere^ if a«y«-|.r*^y = 0, or S^Py = y«03«-.a»). This is the equation of a cone similar and similarly situated to the cone of tangent-lines drawn to the sphere, but its vertex is at the centre. That the equation represents a cone is obvious from the fiict that it is homogeneous in Ty, i. e. that it is in-» dependent of the length of the vector y. [It may be remarked that from the form of the above equa- tion we see that, if x and of be its roots, we have (^7V)(^2V) = a»-^>, which is Euclid, III, 35, 36.] 227* ^nd the locus of the foot of the perpendicular let fall from a given point of a sphere on any tangent-plane. Taking the centre as origin, the equation of any tangent- plane may be written iSWp = o*. The perpendicular must be parallel to p, so that, if we suppose it drawn from the extremity of a (which is a point on the sphere) we have as one value of «r SBCT.228.] THE SPHERE AND CYCLIC CONE. 151 'From these equations^ with the help of that of the sphere we must eliminate p and ^. We have by operating on the vector equation by S.w ©•' = 8a'UT-\'x8vTp •• tsr — a a'C'Bj— a) Hence p = = — -^ — o— • Taking the tensors^ we have the required equation. It may be put in the form and the interpretation of this gives at once a characteristic property of the surface formed by the rotation of the Cardioid about its axis of symmetry. 228, We have seen that a sphere, referred to any point what-* ever as origin, has the equation T{p^a) = 2)3. Hence, to find the rectangle under the segments of a chord drawn through any point, we must put where y is any unit- vector whatever. This gives a?V — 2a?/Say + a* = /3% and the product of the two values of x is i3'— o» This is positive, or the vector-chords are drawn in the same direction, if j^ < jli, i. e. if the origin is outside the sphere. 158 QUATBRNIOKS. [cHAP- VEL 229. AfBwte fixed points ; and, being the origin and P a point in space, find the locus of P, and explain the result when LAOB ia a right, or an obtuse, angle. Let 62 = a, OB = j3, OP = p, then or p*^2S{a+fi)p = -(o«+i8»), or r{p-(a+^)} = V{-28ap). While Soft is negative, that is, while LAOB is acute, the locus is a sphere whose centre has the vector a+jS. It Safi = 0, or LAOB = -, the locus is reduced to the point p = a+i8. IT If Z JO^ >- there is no point which satisfies the conditions. 230. Describe a sphere, with its centre in a given line, so as to pass through a given point and touch a given plane. Let a?a, where a? is an undetermined scalar, be the vector of the centre, r the radius of the sphere, ft the vector of the giv^ point, and Syp =z a the equation of the given plane. The vector perpendicular from the point xa on the given plane is (§208) {a--xSya)Y-\ Hence, to determine x we have the equation T.(a-a?/Sya)>r* = T(xa--^) = r, so that there are, in general, two solutions. It will be a good exercise for the student to find from this equation the condition that there may be no solution, or two coincident ones. 23L Describe a sphere whose centre is in a given line, and which passes through two given points. ^Cf. 232.] THE SPHERE AND CYCLIC CONE. 169 Let the vector of the centre be a? a, as in last section, and let the vectors of the points be p and y. Then, at once. Here there is but one sphere, except in the particular case when we have Ty-Tft, and Say - Sa^, in which case there is an infinite number. The student should carefully compare the results of this section and the last, so as to discover why in general two solutions are possible in the one case, and only one in the other. 232. A sphere touches each of two straight lines, which do not meet : find the locus of its centre. We may take the origin at the middle point of the shortest distance (§ 203) between the given lines, and their equations wiUthenbe p = a+^^, where we have, of course, 5iij8 = 0, /Sa/3i = 0. Let <r be the vector of the centre, p that of any point, of one of the spheres, and r its radius ; its equation is y(p-(r) = r. Since the two given lines are tangents, the following equations in X and x^ must have pairs of equal roots, jr(a-f arj3— a) = r. The equality of the roots in each gives us the conditions Eliminating r we obtain which is the equation of the required locus« 160 QUATEENIONS. [chap. VIL [As we have not^ so far, entered on the consideration of the quaternion form of the equations of the various surfaces of the second order, we may translate this into Cartesian coordinates to find its meaning. If we take coordinate axes of x, y, z respectively parallel to )3, fi^, a, it becomes at once {x-k-myy—iy-k-mxy ^pzy where m and p are constants ; and shows that the locus is a hyperbolic paraboloid. Such transformations^ which are exceed^ ingly simple in all cases, will be of frequent use to the student who is proficient in Cartesian geometry, in the early stages of his study of quaternions. As he acquires a practical knowledge of the new calculus, the need of such assistance will gradually cease to be felt.] Simple as the above solution is, quaternions enable us to give one vastly simpler. For the problem^ may be thus stated-^ Find the locus of the point whose distances from two given lines are equal. And, with the above notation, the equality of the perpendiculars is expressed (§ 201) by yr.(o-(r) up = rr.(o+ (t) up^^ which is easily seen to be equivalent to the equation obtained above. 233« Two spheres being given, show that spheres which cut them at given angles cut at right angles another fixed sphere. If c be the distance between the centres of two spheres whose radii are a and d, the cosine of the angle of intersection is evidently a^^i^^c^ Hence, if a, Oi, and p be the vectors of the centres, and a, a^, r the radii, of the two fixed, and of one of the variable, spheres; A and A^ the angles of intersection, we have (p— a)' + a'-f ^' = 2arcoBAj (p— Oi)*+aJ-f r* = 2airco8Ai. SBOT. 234.] THE SPHERE AND CYCLIC CONE. 161 Elinunating the first power of r, we evidently must obtain a result such as (^ _py + J> + r» = 0, where (by what precedes) p is the vector of the centre, and fi the radius^ of a fixed sphere (p-^)» + i» = 0, which is cut at right angles by all the varjring spheres. By efiTecting the elimination exactly we easily find b and ft in terms of given quantities. 234. To inscribe in a given sphere a closed polygon^ plane or gauche^ whose sides shall be parallel respectively to each of a series of given viectors. Let 2J) = 1 be the sphere^ a, p, y, , rj, the vectors^ n in number^ and let pi, p^y p^ be the vector-radii drawn to the angles of. the polygon. Then pt—Pi = ^i^* &©•> &c. From this, by operating by ^^.(pa -f pi), we get p\^p\ = = i&p, + &pi. Also = Vap^—Vapx. Adding, we get = aps+JTopi = apa-j-pitt. Hence — 1/ Pa=— a-^picu [This might have been written down at once from the result of § 106.] Similarly p, = — j8-»p,/3 = ^^a'^piop, &c. Thus, finally, since the polygon is closed, P«+i = Pi = (-)*a-'ij-» p-'a-'p^ap 71$. We may suppose the tensors of a, j3 ly, ^ to be each unity. Hence, if a = a/3 rjO, Y 162 QITATEBtf IONS. [ohap. VIL we have a-* = ^*iy-' /3-'a~S which is a known quaternion; and thus our condition becomee This divides itself into two cases^ according as n is an eyen or an odd number. If « be even^ we have «Pi = Pi«. Removing the common part p^Sa^ we have This gives one determinate direction^ + Va, for pi ; and shows that there are two^ and only two^ solutions. If » be odd, we have which requires that we have i&::=0. Hence Sap^^ = 0, and therefore p^ may be drawn to any point in the great circle of the unit-sphere whose poles are on the vector a. 235. To illustrate these results, let us take first the case of « = 3. Here we must have S.afiy = 0, or the three given vectors must (as is obvious on other grounds) be parallel to one plane. Here apy, which lies in this plane, is (§ 106) the vector-tangent at the first comer of each of the inscribed triangles ; and is obviously perpendicular to the vector drawn from the centre to that comer. If « = 4, we have PiWr.afiyb, as might have been at once seen from § 106. 236. Hamilton has given {Lectures, p. 674) an ingenious and simple process by which the above investigation is rendered SECT. 238.] THE SPHERE AND CYCLIC CONE. 168 applicable to the more difficult problem in which each side of the inscribed polygon is to pass through a given point instead of being parallel to a given line. His process depends upon the integration of a liuear equation in finite differences. By an immediate application of the linear and vector Amotion of Chapter V, the above solutions may be at once extended to any central surface of the second order. 237. The equation of a cone of revolution, whose vertex is the origin, is easily found. Suppose a, where fa = 1, to be its axis, and e the cosine of its semi- vertical angle ; then, if /> be the vector of any point in the cone, 8aUp = z^e, or S*ap = —e*p*. i 238. Change the origin to the point in the axis whose vector is ;ra, and the equation becomes {—w + SatsrY = — «» (aro^-t!^)^ Let the radius of the section of the coue made by /Sa«r= retain a constant value 6j while x changes ; this necessitates = e. Vb^ + x^ so that when x is infinite, e is unity. In this case the equation becomes ^,„^^^,^.ga ^ 0, which must therefore be the equation of a circular cyliuder of radius b, whose axis is the vector a. To verify this we have only to notice that if cr be the vector of a point of such a cylinder we must (§ 201) have TFaw = b, which is the isame equation as that aboye. Y 2 164 QUATEBNIONS. [cHAP. VIL 289. To find, generaUy^ the equation of a cone which has circular sections : — Take the origin as vertex, and let one of the circular sections be the intersection of the plane 8ap=^ 1 with the sphere (passing through the origin) p« = Spp. These equations may be written thus, SaUp = -^, -Tp = SfiUp. Hence, eliminating Tp, we find the following equation which Up must satisfy — SallpSpUp =-1, or p*—SapSpp = 0, which is therefore the required equation of the cone. As a and fi are similarly involved, the mere /arm of this equation proves the existence of the subcontrary section dis- covered by Apollonius. 240. The equation just obtained may be written or, since a and j3 are perpendicular to the cyclic arcs (§ 69*), sinjosiny= constant, where p and y are arcs drawn from any point of a spherical conic perpendicular to the cyclic arcs. This is a well-known property of such curves. 24L If we cut the cyclic cone by any plane passing through the origin, as Syp = 0, then Fay and Ffiy are the traces on the cyclic planes, so that p = xUFay+yUFpy (§ 29). SECT. 244.] THE SPHBBE AND CYCLIC CONE. 165 Substitute in the equation of the cone^ and we get where P is a known scalar. Hence the values of x and y are the same pair of numbers. This is a very elementary proof of the proposition in § 69*, that PL = MQ (in the last figure of that section). 242. When x and y are equals the transversal arc becomes a tangent to the spherical conic, aild is evidently bisected at the point of contact. Here we have This is the equation of the cone whose sides are perpendiculars (through the origin) to the planes which touch the cyclic cone. 243. It may be well to observe that the property of the Stereographic 'projection of the sphere, viz. that the projection of a circle is a circle, is an immediate consequence of the above form of the equation of a cyclic cone, 244. That § 239 gives the most general form of the equation of a cone of the second order, when the vertex is taken as origin, follows from the early results of next Chapter. For it is shown in § 249 that the equation of a cone of the second order can always be put in the form 2'2,.8ap8^p^Ap^ = 0. This may be written 8p<i>p = 0, where ^ is the self-conjugate linear and vector function <^p = 2r.apj3+(^ + 25ai3)p. By § 168 this may be transformed to and the general equation of the cone becomes (P'^S\fi)p^-\.28\p3fip = 0, which is the form obtained in § 239» 166 QUATEENIONS. [cHAP. VIL 245. Taking the form 8p<l>p = as the simplest^ we find by differentiation Sdp<l>p+8pd<f>p = 0, or 28dp^p = 0. Hence ^p is perpendicular to the tangent-plane at the extremity of p. The equation of this plane is therefore (^ being the vector of any point in it) or^ by the equation of the oone^ flW^p = 0. 246. The equation of the cone of normals to the tangent- planes of the given cone can be easily formed from that of the cone itself. For we may write it in the form «(*"'#)# = 0, and if we put ^p = o-^ a vector of the new cone^ the equation becomes Saifr^tT = 0. Numerous curious properties of these connected cones^ and of the corresponding spherical conies^ follow at once from these equations. But we must leave them to the reader. 247. As a final example^ let us find the equation of a cyclic cone when five of its vector-sides are given — ^i. e. find the cone of the second order whose vertex is the origin, and on whose surface lie the vectors a, j3, y, 5> €. If we write = S.r{rapn€)F(FpyF€p)r(FybVpa), (1) we have the equation of a cone whose vertex is the origin — ^for the equation is not altered by putting xp for p. Also it is the equation of a cone of the second degree, since p occurs only twice. Moreover the vectors o, fi, y, 5, c are sides of the cone, CHAP. Vn.] EXAMPLES TO CHAP. VII. 167 because if any one of them be put for p the equation is satisfied. Thus if we put fi for p the equation becomes = S.FiFapn^) FiFfiyFtP) F(FybFpa) :=S.F{FapFh€){Ffia8.FybFfiyF€fi-'Fyb8.FpaFpyF€p]. The first term vanishes because S.F{FapFh€)Ffia=zO, and the second because S.F^aF^yF^fi := 0, since the three vectors F^a, F^y, Fcft being each at right angles to fiy must be in one plane. As is remarked by Hamilton^ this is a very simple proof of Fascal^s Theorem — ^for (1) is the condition that the intersections of the planes of a^)3 and d^c; fi,y and €,p; y,h and p^a; shall lie in one plane ; or^ making the statement for any plane section of the cone, the points of intersection of the three pairs of opposite sides, of a hexagon inscribed in a conic, lie in one straight line. EXAMPLES TO CHAPTER VII. /^ 1 . On the vector of a point P in the plane Sap = I a point Q is taken, such that QO.OP is constant; find the equation of the locus of Q. 2. What spheres cut the loci of P and Q in (1) so that both lines of intersection lie on a cone whose vertex is ? 3. A sphere touches a fixed plane, and cuts a fixed sphere. If the point of contact with the plane be given, the plane of the intersection of the spheres contains a fixed line. 168 QUATEBNIONS. [cfHAP. VII Find the locos of the centre of the variable sphere^ if the plane of its intersection with the fixed sphere passes through a given point. 4. Find the radii of the spheres which touchy simultaneonsly, the four given planes iSap = 0, SppszO, 8yp = 0, 58p=l. [What is the volume of the tetrahedron enclosed hj these planes?] 5. If a moveable line^ passing through the origin, make with any number of fixed lines angles 0, 0^, 0^, &c«^ such that acos.d-h«iCOsA + = constant, where a, a^, are constant scalars, the line describes a right cone.* 6. Determine the conditions that may represent a right cone. 7. What property of a cone (or of a spherical conic) is given directly by the following form of its equation, 8.1PKP = ? 8. What are the conditions that the surfaces represented by 8p(ftp = 0, and Aipxp = 0, may degenerate into pairs of planes ? 9. Find the locus of the vertices of all right cones which have a common ellipse as base. 10. Two right circular cones have their axes parallel, show that the orthogonal projection of their curve of intersection on the plane containing their axes is a parabola. 11. Two spheres being given in magnitude and position, every sphere which intersects them in given angles will touch two other fixed spheres and cut another at right angles. CHAP. Vn.] EXAMPLES TO CHAPTER VII. 16& 1 2. K a sphere be placed on a table^ the breadth of the elliptic shadow formed by rays diverging from a fixed point is inde- pendent of the position of the sphere. 13. Form the equation of the cylinder which has a given circular section^ and a given axis. Find the direction of the normal to the subcontrary section. 14. Given the base of a spherical triangle, and the product of the cosines of the sides, the locus of the vertex is a spherical conic, the poles of whose cyclic arcs are the extremities of the given base. 15. (Hamilton, Bishop Lavfs Premium Ex., 1858.) {a.) What property of a sphero-conic is most immediately indicated by the equation a p (J.) The equation (rAp)« + (%>)» = also represents a cone of the second order; X is a focal line, and pi is perpendicular to the director-plane cor- responding. (c.) What properly of a sphero-conic does the equation most immediately indicate ? 16. Show that the areas of all triangles, bounded by a tangent to a spherical conic and the cyclic arcs, are equal 17. Show that the locus of a point, the sum of whose arcual distances &om two given points on a sphere is constant, is a spherical conic. 18. If two tangent planes be drawn to a cyclic cone, the four lines in which they intersect the cyclic planes are sides of a right cone. z 170 QUATBRBTIONS. [cHAP. VII. 19. Find the equation of the cone whose sides are the in- tersections of pairs of perpendicular tangent planes to a giyen cyclic cone. 20. Find the condition that five given points may lie on a sphere. 21. What is the surface denoted by the equation where p = xa-^yfi-^zy, a, p, y being g^ven vectors, and x^ y, z variable scalars ? Express the equation of the surface in terms of py a, p, y alone. 22. Find the equation of the cone whose sides bisect the angles between a fixed line and any line, in a given plane, which meets the fixed line. What property of a spherical conic is most directly given by this result ? CHAPTER VIII. SURFACES OF THE SECOND ORDER. 248. npHE general scalar equation of the second order -*• in a vector p must evidently contain a term in- dependent of p, terms of the form SMpb involving p to the first degree, and others of the form 8.apbpc involving p to the second degree, «, b, c, &c. being constant quaternions. Now the term 8.apb may be written 8.{8a^'ra)p{Sb^n), or 8a8prb^8b8pra^8.pfbray each of which may evidently be put in the form 8yp, where y is a known vector. Similarly the term 8.apbpc may be reduced to a set of terms, each of which has one of the forms Ap\ {8apy, Sap8pp, the second being merely a particular case of the third. Thus (the numerical factors 2 being introduced for convenience) we may write the general scalar equation of the second degree as follows : — 22./Sap/Sl3p+^p'+2%= C. (1) 249. Change the origin to D where OD = 5, then p becomes p-^by and the equation takes the form 22.8ap8pp-^Ap^ + 2:^{8ap8fib+8pp8ab)'^2A8dp + 28yp + 22.8ab8pb+Ab'-i'28yb-C=z 0; from which the first power of p disappears, that is tie surface is referred to its centre, if ^{a8p»-^p8ab)i-Ab + y = 0, (2) 172 QUATERNIONS. [chap.VIIL a vector equation of the first degree^ which in general gives a single definite value for b, by the processes of Chapter V. [It would lead us beyond the limits of an elementary treatise to consider the special cases in which (2) represents a line, or a plane, any point of which is a centre of the surface. The processes to be employed in such special cases have been amply illustrated in the Chapter referred to.] With this value of b, and putting the equation becomes 22.&vfi)3p+^« = i). If i^=0, the surface is conical (a case treated in last Chapter); if not, it is an ellipsoid or hyperboloid. Unless expressly stated not to be, the surface will, when D is not zero, be considered an ellipsoid. By this we avoid for the time some rather delicate considerations. By dividing by D, and thus altering only the tensors of the constants, we see that the equation of central surfaces of the second order, referred to the centre, is (excluding cones) 21{8apSfip)+ffp' =^ I (3) 250. Differentiating, we obtain 22{8adp8pp + 8apSpdp} -^2ff8pdp = 0, or 8.dp{^{a8pp-^p8ap)+ffp} = 0, and therefore, by § 137, the tangent plane is 8{vr^p){l{a8pp + p8ap)+ffp} = 0, i.e. 8.^{2{a8fip+p8ap)-h^p} = 1, by (3). Hence if v = 2{a8pp-\-p8ap)+ffp (4) the tangent plane is 81;^ = 1, and the surface itself is 8vp = 1. And, as y-* is evidently the vector-perpendicular from the origin on the tangent plane, v is called the vector of proximity. SECT. 252.] SUBPACBS OP THE SECOND OBDER. 173 25L Hamilton uses for v, which is obviously a linear and vector function of p, the notation (pp, (p expressing a functional operation^ as in Chapter V. But, for the sake of clearness, we will go over part of the ground again, especiallj for the benefit of students who have mastered only the more elementary parts of that Chapter. We have, then. With this definition of ^, it is easy to see that (a.) ^(p-f-cr) = <l>p + <l><T, &c., for any two or more vectors. (6.) <p{xp) = X(f>py a particular case of (a), x being a scalar. {c.) d<l>p = 4>{dp). (d.) Scrimp = 1 {Sa<TSpp + Sfi(r Sap) -^ff Spar = Sptpar, or <^ is, in this case, self-conjugate. This last property is of great importance. 252. Thus the general equation of central surfaces of the second degree (excluding cones) may now be written Sp<l>p^l (1) Differentiating, Sdp<f>p +'Spd<t>p = 0, which, by applying (c.) and then {d.) to the last term on the left, gives 2S<t>pdp=. 0, and therefore, as in § 250, though now much more simply, the tangent plane at the extremity of p is /S(tsr-p)<^p = 0, or SmiPp = Sp<f}p = 1. K this pass through A{OA = a), we have Sa<l>p = 1> or, by (d.), Spi^a = 1, for all possible points of contact. 174 QUATERNIONS. [CHAP-VIIL This id therefore the equation of the plane of contact of tangent planes drawn from A, 253. To find the enveloping cone whose vertex is A, notice ttat {Sf4p^ 1) +i>(5>)<^a- \y = 0, where j9 is any scalar^ is the equation of a surface of the second order touching the ellipsoid along its intersection with the plane. If this pass through A we have andj9 is found. Then our equation becomes (S/M(>p-l)(/S»<^a-l)-(/S^-l)« = 0, (1) which is the cone required. To assure ourselves of this, transfer the origin to A, by putting p + a for p. The result is, using {a.) and {d.\ {Sp<l>p + 28p<l>a-i'8a4>a-l){Sa4>a^l)-{8p4>a-i'Sa4>a- 1)» = 0, or Sp<l)p{Sa(t>a — 1 ) - {Sp<t>ay = 0, which is homogeneous in Tp^, and is therefore the equation of a cone. Suppose A infinitely distant, then we may put in (1) ^a for a, where x is infinitely great, and, omitting all but the higher terms, the equation of the cylinder formed by tangent lines parallel to a is (SJj^p- 1) Sa<l)a-{Sp<l>ay = 0. 254. To study the nature of the surface more closely, let us find the locus of the middle points of a system of parallel chords. Let them be parallel to a, then, if «r be the vector of the middle point of one of them, 'ur-^-xa and 'ur-^xa are simultaneous values of p which ought to satisfy (1) of § 252. That is 8.{^±xa)4>{:fir±xa) = 1. Hence, by {a.) and (rf.), as before, Sm(fyGr + x^ Sa<t>a =1, Sw<t>a=0 (1) »BCT. 255.] SURFACES OF THE SECOND ORDER. 175 The latter equation shows that the locus of the extremity of 'Gj^ the middle point of a chord parallel to a^ is a plane through the centre, whose normal is ^a; that is, a plane parallel to the tangent plane at the point where OA cuts the surface. And (d.) shows that this relation is reciprocal — so that if 13 he any value of -or, i.e. be any vector in the plane (1), o will be a vector in a diametral plane which bisects all chords parallel to ^. The equations of these planes are Sxa^a = 0, &fs^^ = 0, so that if F.ffxuf)^ = y (suppose) is their line of intersection, we ^^^^ Sycl>a = = 8a4>y ' Sy(l>fi =z = Sp(l>Y ►, (2) and (1) gives Sp(l>a = = iSo<^/3 . Hence there is an infinite number of sets of three vectors <^y Py yy ^^^ ^^^ ^^^ chords parallel to any one are bisected by the diametral plane containing the other two, 255. It is evident from § 23 that any vector may be ex- pressed as a linear function of any three others not in the same plane, let then p=^xa-{-yp-\-zy, where, by last section, 8a<i>p = S^(i>a = 0, Sai^y =: Sy<l)a =^ 0, 8fi(l>y = 8y(l>fi = 0. And let . iSou^a = 1 8^P = 1 Sy(l>y = 1 J SO that a, /3, and y are vector conjugate semi-diameters of the surface we are engaged on. Substituting the above value of p in the equation of the 176 QUATRRNIONa [CHAF.YIIL sur&oe^ and attending to the equations in.a,Pfy and to {a.), {b.), and (d.), we have To transform this equation to Cartesian coordinates^ we notice that X is the ratio which the projection of p on a bears to a itself^ &c. If therefore we take the conjugate diameters as axes of ^, fi, Cf <uid their lengths as a, b, c, the above equation becomes at once ^. + i. + ^. - ^> the ordinary equation of the ellipsoid referred to conjugate diameters. 256. If we write — ^* instead of ^, these equations assume an interesting form. We take for granted^ what we shall after- wards prove^ that this halving or extracting the root of the vector function is lawful^ and that the new linear and vector function has the same properties {a.\ (b.), {c.)^ {d.) (§ 251) as the old* The equation of the surface now becomes «^v=-i, or S'^p^p = — 1, or, finally, 7\^p = 1. If we compare this with the equation of the unit-sphere we see at once the analogy between the two sur&ces. The ^here can be changed into the ellipsoid, or vice versa, by a linear deformation of each vector, the operator being the function \^ or its inverse. See the Chapter on Physical Applications. 257. Equations (2) § 254 now become iSa\^«/3 = = SyjfaylfP, fee, (1) so that ^a, ^)3, yfry, the vectors of the unit^sphere which cor- SECT. 258.] SURFACES OF THE SECOND ORDER. 177 respond to aemUconjtigate diameters of the ellipsoidy form a rect- angular system. We may remark here, that, as the equation of the ellipsoid referred to its principal axes is a case of § 256, we may now suppose i,y, and k to have these directions, and the equation is J?* y* ^» _ . _ . "Y + ^ ,+ ^ = 1> which, m quaternions, is a* d' c* We here tacitly assume the~ existence of such axes, but in all cases, by the help of Hamilton's method, developed in Chapter V, we at once arrive at the cubic equation which gives them. It is evident from the last- written equation that A.. . *^^*^ _^j8jp kSkp and ^p^^C^ +-^^ + ^, ^ a b c ^ which latter may be easily proved by showing that And this expression enables us to verify the assertion of last section about the properties of V^. As Sip = — a?, &c., X, y, z being the Cartesian coordinates referred to the principal axes, we have now the means of at once transforming any quaternion result connected with the ellipsoid into the prdinary one. 258. Before proceeding to other forms of the equation of the ellipsoid, we may use those already given in solving a few problems. Find the loeus of a point when the perpendicular Jrom the centre on its polar plane is of constant length. If «r be the vector of the point, the polar plane is iS|p<^'cr =1, and the length of the perpendicular from is ijrrz (§ 205). A a 178 QUATERNIONS. [cHAP.VIII. Hence the required locus is or Svr<l>*vr^—C*, a concentric ellipsoidj with its axes in the same direction as those of the first. By § 257 its Cartesian equation is ^* + 5* + ^i - ^ • 258. M^ the locus of a point whose distcmcefrom a given point is ahoays in a given ratio to its distance front a given line. Let /j=a?)3 be the given line, and A (OA = a) the given point, and let Sa^ = 0. Then for any one of the required points T{p^a) = eTrfip, a surface of the second order, which may be written p«-2iSap+a« = e^{8^pp-^PY). Let the centre be at d, and make it the origin, then p' + 28p{b^a)+{b-ay = e' {S\p{p+b)^P'{p + by], and, that the first power of p may disappear, (d-a) = ^»(/3/Sl35-i3^8), a linear equation for S. To solve it, note that Sa^ = 0, operate by 8.p and we get (1 -6«/3« + 6«/3»)S>3« = Spb = 0. Hence 5 — a = — e^fi% or 8 = 1+e^fi^ Referred to this point as origin the equation becomes which shows that it belongs to a surface of revolution whose axis is parallel to ^, as its intersection with a plane Spp = a, perpendicular to that axis, lies also on the sphere SECT. 2G1.] SUBFACES OP THE SECOND OBDBB. 179 260. A sphere, passing through the centre of an ellipsoid^ is cut by a series of spheres whose centres are on the ellipsoid and which pass through the centre thereof $ Jmd the envelop of the planes of intersection. Let (/}— o)* = a' be the first sphere, i.e. p«-2^ap = 0. One of the others is • p*— 2i8Wp=0, where &Gr<^^ 1. The plane of intersection is 8{'ar^a)p = 0. Hence, for the envelop, (see next Chapter,) Sv/4m=: ' f where m^z Srs/p^ 0, i or ^ = xp, {Vx^ = 0}, i. e. tsr = x<fr^p. Hence a?«6'/)0-V = 1, Sap,' and x8p<tr^p:=z and, eliminating x, Spi^'-p = {8ap)\ a cone of the second order. 26L From a point in the outer of two concentric ellipsoids a tangent cone is drawn to the inner, find the envelop of the plane of contact. If Sur<pw=z 1 be the outer, and 8p\lrp = 1 be the inner, 4> and yjr being any two self-conjugate linear and vector functions, the plane of contact is Svnjrp^ 1. Hence, for the envelop, Sv/yjrp = 0, "i iSW'^ = 0, J therefore (fm = xyjrp, or -or = Xfpr^y^p, A a 2 180 QUATBRNIOKS. [cHAP,VnL This g^ves xS.y^pf^-^y^p = 1^ > and x*S.y^p4>''hlrp = 1, ) and thereforej eliminating x, 8.ylfp<t>-'y^p = 1, or S,p\lf<l>-*\lfp = 1, another concentric ellipsoid^ as ^<^-^^ is a linear and vector Amotion = x suppose ; so that the equatioi^ may be written Spxp = 1. 262. Find tie locus of intersection of tangent planes at tie extremities of conjugate diameters, U a, fi,y be the vector semi-diameters^ the planes are Swyjf^a = — 1, with the conditions § 257. Hence — ^S.^a^^^i/ry = ^ = ^a+^I^P-^ylryj by § 92, therefore ^-bt = \/3, since yjfay ^j9, ^y form a rectangular system of unit- vectors. This may also evidently be written Smjf^'Gr =—3, showing that the locus is similar and similarly situated to the given ellipsoid, but larger in the ratio \/3 : 1. 263. Find the locus of the intersection of three sphere whm diameters are semi-conjugate diameters of an ellipsoid. If a be one of the semi-conjugate diameters And the corresponding sphere is p'— /Sap = 0, or p" — iiS\/ra\/f-*p = 0, SECT. 264.] SURFACES OF THE SECOND ORDER. 181 with similar equations in p and y. Henoe^ by § 92^ and^ taking tensors^ or JV^-V"' = v^3, or, finally, Sp^~*P = — 3 p*. This is Fresnel^s Surface of Elasticity in the Undulatory Theory. 264. Before going farther we may prove some useful proper- ties of the function ^ in the form we are at present using — viz. iSip jSjp kSkp We have p = ^iSip^jSjp^kSkp, and it is evident that i ^- J ,7 * Hence _ ^'^^P J'^P ^^^P ^-»p = aH8ip+b^J8;p-^e*kSkp, Also and so on. Again, if a, jS, y be any rectangular imit- vectors oa9«= ^, +-^ -r ' — —} &c. = &c. But as (5ip)' + (^p)' + (%)'=-p', we have Sa>t>a+SP4,p+Sy4^ = J. + J_ + i.. Again, 8.ii^P4>y = ^•(^+-)(^'+.-.)C+-) Sia a* ' Sip a» ' Siy a" ' b^ ' c« b- ' c- Sjy Sky b* ' c^ — 1 "■ a^b^c^ Sia, Sja, Ska Sip, Sjp, Skp Siy, Sjy, Sky 1 182 QUATERNIONS. [cHAP.VIIL And so on. These elementary investigations are given here for the benefit of those who have not read Chapter Y. The student may easily obtain all such results in a far more simple maimer by means of the formula of that Chapter. 265. Find the locus of intersection of a rectangular system of three tangents to an ellipsoid. K cr be the vector of the point of intersection, o, ft y the tangents, then, since v-\-xa should give equal values of or when substituted in the equation of the surface, giving i8'(or4-^a)^(«r4'iPa) = 1, or x^8a4>a + 2x8ml>a+ (5W^— 1) = 0, we have (5W^)* = &i^(5Br<^— 1). Adding this to the two similar equations in ft and y {8a4^y + iSfi<hry + (Sy<K)* = iSo4>a + 8p<l>p + Syi^y) {STfft^^ 1), or -(^)« = (^ + 1- + ^) (&r<^-l), an ellipsoid concentric with the first. 266. If a rectangular system of chords be drattm through any point within an ellipsoid^ the sum of the reciprocals of the rect- angles under the segments into which they a/re divided is constant. With the notation of the solution of the preceding problem, «r giving the intersection of the vectors, it is evident that the product of the values of x is one of the rectangles in question taken negatively. Hence the required sum is _L i- i. 2&w^a g« "^ g' "^ C Smiftrff — 1 S'sr^'sr — 1 This evidently depends on Server only and not on the par- SECT. 267.] SURFACES OF THE SECOND ORDEB. 183 ticular directions of a,p,y: and is therefore unaltered if 'cf be the vector of any point of an ellipsoid similar^ and similarly situated, to the given one. [The expression is interpretable even if the point be exterior to the ellipsoid.] 267. Show that if any rectangular system of three vectors he drawn from a point of an ellipsoid, the plane containing their other extremities passes through a fixed point. Find the locus of the latter point as the former varies. With the same notation as before, we have SxiTifyGr = 1, and 8{sr-^xa)(l>{vr+xa) = 1 ; therefore a? = ^ , . ooupa Hence the required plane passes through the extremity of and those of two other vectors similarly determined. It therefore passes through the point whose vector is aSa(f>zr + pSp(lm + ySytfm (9 = tsr— 2 Swl>a-\'Spit>p + SY(t>y or (9 = tsr+ ^^ (§ 173). m^ ^^ ^ Thus the first part of the proposition is proved. But we have also -=?(♦+?)"'». whence by the equation of the ellipsoid the equation of a concentric ellipsoid. 184 QUATBKNIONS. [cHAP.Vm. 268. Find tie directions of the three vectors wAicA are parallel to a set of conjugate diameters in eacA of two central surfaces of tAe second degree^ Transferring the centres of both to the origin, let their equations be 8(4p = 1 or 0, ^ .J. and Spyirp = 1 or 0. i ^ If o, /3, y be vectors in the required directions, we must have 8a(l>p = 0, Sayjffi = 0, ^ 8p<l>y^0, Spylry==0, ^ (2) Syifya = 0, iSy^o = 0. ^ From these equations <l>a\\rpyUa, &c. Hence the three required directions are the roots of r.<^/Wrp = (3) This is evident on other grounds, for it means that if one of the surfaces expand or contract uniformly till it meets the other , it will touch it successively at points on the three sought vectors. We may put (3) in either of the following forms — or r.py^^ii^p = 0,J and, as ^ and i/r are given functions, we find the solutions by the processes of Chapter V. [Note. As <fr^y\r and ^-*0 are not, in general, self-conjugate functions, equations (4) do not signify that a, ft y are vectors parallel to the principal axes of the surfaces -^#=1, 5.pV^-^^p=l.] 269* Find the equation of the ellipsoid of which three con- jugate semi-diameters are given. (SHBOT. 270.] SUBFACES OF THE SECOND ORDEB. 185 Let the vector semi-diameters be a, p, y, and let 8p<l>p s= 1 be the equation of the ellipsoid. Then (§ 255) we have ScuJHi = 1, Sa4>p = 0, Syijyy = 1, Sy<l>a = ; the six scalar conditions requisite (§ 139) for the determination of the linear and vector function 0. They give a || r<^/3</)y, or xa = <l>~^ Fpy» Hence x = xScuIm = S.afiy, and similarly for the other combinations. Thus^ as we have pS.apy = aS.pyp-^pS.yap+yS.app, we find at once 4>pS*.aPy = FpyS.pyp+ VyaS.yap-^' FafiS.oifip ; and the required equation may be put in the form SKapy = SKapp'\-SKfiyp + S*.yap. The immediate interpretation is that if four tetrahedra he formed hy grottping, three and three, a set of semi-conjugate vector axes of an ellipsoid and any other vector of the surface, the sum of the squares of the volumes of three of these tetrahedra is equal to the square of the volume of the fourth. 270. When the equation of a surface of the second order can be put in the form ^p</>-V=l, (1) where (</>-y) (</>— ^i) (</>— y>) = 0, we know that g, g^^g^ are the squares of the principal semi- diameters. Hence, if we put </>-|-A for </> we have a second surface, the differences of the squares of whose principal semi- axes are the same as for the first. That is, ^p(*-h*)-V=l (2) Bb 186 QUATBENIONa [OHAP.VIIL is a surface canfocal with (1). From this simple modification of the equation all the properties of a series of confocal surfaces may easily be deduced. We give one as an example. 27L Any two confocal suffaces of the second order , whkb meet, intersect at right angles. For the normal to (2) is, evidently, and that to another of the series, if it passes through the common point whose vector is p, is there = j^^ «p((0+ao-*-(«+A)-Op> and this evidently vanishes if h and h^ are difiPerent, as they must be unless the surfaces are identical. 272. To find the conditions of similarity of two central surfaces of the second order, Referring them to their centres, let their equaitions be i ^^P-'>\ ^1) Now the obvious conditions are that the axes of the one are proportional to those of the other. Hence, if g' -^m^g^ + mig '{■m^0,'\ /' + </" + </+»*'= 0, J be the equations for determining the squares of the reciprocals of the semiaxes, we must have -:t- = M> -z- = M% — = f^> v3) m^ Ml m |vbere fi is an undetermined scalar* Thus it appears that there I :l\ ■■ ™ SECT. 273.] SURFACES OF THE SECOND ORDER. 1S7 are but two scalar conditions necessary. Eliminating fx we have flll\ M^ VIM2 fill which are equivalent to the ordinary conditions. 273. Find the greatest and least semi-diameters of a central plane section of an ellipsoid. Here iSp</>p = 1 Sap together represent the elliptic section; and our additional coU"> dition is that ^ is a maximum or minimum. Differentiating the equations of the ellipse, we have /S^prfp = 0, Sddp = 0, and the maximum condition gives dTp = 0, or Spdp = 0. Eliminating the indeterminate vector dp we have 8,ap4>p=iO (2) ' This shows that tAe mammum or minimum vector, the normal at its extremity y and the perpendicular to the plane of section, lie in one plane. It also shows that there are but two vectors which satisfy the conditions, and that they are perpendicular to each other, for (2) is satisfied if ap be substituted for p. We have now to solve the three equations (1) and (2), to find the vectors of the two (four) points in which the ellipse (1) intersects the cone (2). We obtain at once ^p =a xF.ffr^aFap. Operating by S.p we have 1 = xp^8a^-^a. B b 2 188 QlTATB]tNIOK& [chap.VIII. Hence p.^p^^p^a^l^ fromwhich &o(l-p*^)-*o = 0; (4) a quadratic equation in p*, from which the lengths of the maximum and minimum vectors are to be determined. By § 147 it may be written «p*&i^-»o-p«-S.o(«,-*)o+a« = (5) [If we had operated by S.<l>-^a instead of by S.a, we should have obtained an equation apparently different from this^ but easily reducible to it. To prove their identity is a good exercise for the student.] Substituting the values of p* given by (5) in (3) we obtain the versors of the required diameters. [The student may easily prove directly that (l-pl^)-»a and (l-pj*)-*a are necessarily perpendicular to each other^ if both be perpen- dicular to a, and if pj and pi be different.] 274. By (4) of last section we see that Hence the area of the ellipse (1) is vTa Also the locus of normals to all diametral sections of an ellipsoid, whose areas are equal, is the cone 8(j4-^a = (7o». When the roots of (4) are equal, i. e. when (««,a*-^5a0a)^ = 4 ma*Sa(l>'% (5) the section is a circle. It is not di£Scult to prove that this SECT. 277.] SUEPACBS OP THE SECOND ORDER. 189 equation is satisfied by only two values of Ua, but another quaternion form of the equation gives the solution of this and similar problems by inspection. 275. By § 168 we may write the equation 8p<t>p = 1 in the new form 8.kpfip-\-pp* = 1, where /? is a known scalar^ and X and fx are definitely known (with the exception of their tensors^ whose product alone is given) in terms of the constants involved in (f>, [The reader is referred again also to §§ 121^ 122.] This may be written 28KpSixp'{-{p^8KiJL)p^ = 1 (1) From this form it is obvious that the surface is cut by any plane perpendicular to X or m in a circle, For^ if we put 8\p = a, we have 2a8p.p'{'{p^8\p)p^ = 1, the equation of a sphere which passes through the plane curve of intersection. Hence X and fi of § 168 are^ the values of a in equation (5) of last section. 276. -4«y two circular sections of a central surface of the second order J whose planes a/re not parallel^ lie on a sphere. For the equation {8Kp—a){8p.p'^h) = 0, where a and h are any scalar constants whatever^ is that of a system of two non-parallel planes^ cutting the surface in circles. Eliminating the product 8Kp8[kp between this and equation (1) of last section^ there remains the equation of a sphere. 277. To find the generating lines of a central surface of the second order, . Let the equation be 8p<\>p = 1 ; 190 QUATBBNIONS. [ohap.VIIL then^ if a be the vector of any })omt on the surface^ and 9 a vector parallel to a generating line^ we must have p =: a+xvr for all values of the scalar x. Hence iS(o+a?cr)^(a-f^ = 1, which gives the two equations Saifyer = 0, ^ Sm<l>vr = 0. J The first is the equation of a plane through the origin parallel to the tangent plane at the extremity of a, the second is the equation of the asymptotic cone. The generating lines are therefore parallel to the intersections of these two surfaces^ as is well known. From these equations we have where ^ is a scalar to be determined. Operating on this by &j9 and S.y, where /3 and y are any two vectors not coplanar with a, we have StsT(y(t>P+ Vap) = 0, Sm{y<l>y- Vya) = (1) Hence S.tf^iyi^fi + ?^a/3)0</)y- Vya) = 0, or my^S.aPy-'ScupaS.afiy = 0. Thus we have the two values belonging to the two generating lines. 278. But by equation (1) we have Zfsr = F:(y</)^+ raP){y4>y- Tya) = my^<t>-^rpy+yF.<l>arpy''aS.aTpY; which^ according to the sign of y, gives one or other generating line. Here Fpy may be any vector whatever, provided it is not .SECT. 278.] SURFACES OF THE SECOND OBDEB. 19J perpendicular to a (a condition assumed in last section)^ and we may write for it $. Substituting the value of y before founds we have = r.<t>ar.o4-'0±^^2^r,l>ae. 8afl>a m or, as we may evidently write it^ ^(tr'{r.ar<l>ae)±y/^F4>a0 (2) Put r = r<i>ae, and we have ,8^ ziff = dr^Var + </ — ^— r, with the condition 8r^ = 0. [Any one of these sets of values forms the complete solution of the problem; but more than one have been given^ on account of their singular nature and the many properties of surfaces of the second order which immediately follow from them. It will be excellent practice for the student to show that yffO = U{r4ara<i>''0 ± ^ ^ V^ai) is an invariant. This may most easily be done by proving that V.y^Q^By^ = identically.] Perhaps^ however, it is simpler to write a for F./Sy, and we thus i)btain 2ftsr = </)-' r,a ra4>a ± ^ — ^ Vw^a, [The reader need hardly be reminded that we are dealing with the general equation of the central surfaces of the second order — ^the centre being origin.] 192 QUATBRNIOira [cHAP.Vni. EXAMPLES TO CHAPTER VHI. !• Find the locus of points on the surface where the generating lines are at right angles to one another. 2. Find the equation of the surface described by a straight line which revolves about an axis^ which it does not meet but^ with which it is rigidly connected. 3. Find the conditions that may be a sur&ce of revolution. 4. Find the equations of the right cylinders which dream- scribe a given ellipsoid. 5. Find the equation of the locus of the extremities of per- pendiculars to plane sections of an ellipsoid^ erected at the centre^ their lengths being the principal semi-axes of the sec- tions. [Fresnd^s Wave'Surface.'] 6. The cone touching central plane sections of an ellipsoid^ which are of equal area, is asymptotic to a confocal hyperboIoid« 7. Find the envelop of all non-central plane sections of an ellipsoid whose area is constant. 8. Find the locus of the intersection of three planes, per- pendicular to each other, and touching, respectivdy, each of three confocal surfaces of the second order. 9. Find the locus of the foot of the perpendicular £rom the centre of an ellipsoid upon the plane passing through the ex- tremities of a set of conjugate diameters. CHAP. VIII.] EXAMPLES TO CHAP. VIII. 193 10. Find the points in an ellipsoid where the inclination of the normal to the radius- vector is greatest. 11. If four similar and similarly situated surfaces of the second order intersect, the planes of intersection of each pair pass through a common point. 12. If a parallelepiped be inscribed in a central surface of the second degree its edges are parallel to a system of conjugate diameters. 13. Show that there is an infinite number of sets of axes for which the Cartesian equation of an ellipsoid becomes 14. Find the equation of the surface of the second order which circumscribes a given tetrahedron so that the tangent plane at each angular point is parallel to the opposite face ; and show that its centre is the mean point of the tetrahedron. 15. Two similar and similarly situated surfaces of the second order intersect in a plane curve, whose plane is conjugate to the vector joining their centres. 16. Find the locus of all points on SfKpp = 1, where the normals meet the normal at a given point. Also the locus of points on the surface, the normals at which meet a given line in space. 17. Normals drawn at points situated on a generating line are parallel to a fixed plane. 18. Find th^ envelop of the planes of contact of tangent planes drawn to an ellipsoid from points of a concentric sphere. Find the locus of the point from which the tangent planes are drawn if the envelop of the planes of contact is a sphere. 19. The sum of the reciprocals of the squares of the per- pendiculars from the centre upon three conjugate tangent planes is constant. c c 194 QUATBENIONS. [chap.VIIL 20. Cones are drawn^ touching an ellipsoid^ from any two points of a similar^ similarly situated^ and concentric ellipsoid* Show that they intersect in two plane curves. Find the locus of the vertices of the cones when these plane sections are at right angles to one another. 21. Find the locus of the points of contact of tangent planes which are equidistant from the centre of a surface of the second order. 22. From a fixed point A, on the surface of a given sphere, draw any chord AD; let 1/ be the second point of intersection of the sphere with the secant BD drawn from any point B; and take a radius vector AE, equal in length to BI/, and in direc- tion either coincident with^ or opposite to, the chord AB : the locus of ^ is an ellipsoid, whose centre is A, and which passes through B. (Hamilton, Elements, p. 227.) 23. Show that the equation l*{e*^\){e'\'8aa) = (&ip)«-2<?iSiip-8'aV-f(&i»>-|-(l-dV. where e is a variable (scalar) parameter, represents a system of confocal surfaces. {Ibid, p. 644.) 24. Show that the locus of the diameters of which are parallel to the chords bisected by the tangent planes to the cone Spy^p = is the cone -8'.p0t^-*<^p = 0. 25. Find the equation of a cone, whose vertex is one summit of a given tetrahedron, and which passes through the circle circumscribing the opposite side. 26. Show that the locus of points on the surface = 1, CHAP. VIIL] EXAMPLES TO CHAP. VIII. 196 the normals at wliich meet that drawn at the point p = «r^ is on the cone 27. Find the equation of the locus of a point the square of whose distance from a given line is proportional to its distance from a given plane. 28. Sho^ that the locus of the pole of the plane Sap=^ 1, wiUi respect to the surface 8p4>p = 1, is a sphere^ if a be subject to the condition 29. Show that the equation of the surface generated by lines drawn through the origin parallel to the normals to Spip'^p = 1 along its lines of intersection with Sp{<l> + i)-'p = 1, is «r«—M«r (</>+*)-* tsr = 0. 30. Common tangent planes are drawn to 2S\p8iip-^{P'-8Xp)p^ = 1, and Tp = A, find the value of A that the lines of contact with the former surface may be plane curves. What are they^ in this case^ on the sphere ? 31. If tangent cones be drawn to Sp^P = 1, from every point of 8p(l>p = 1, the envelop of their planes of contact is 8p<l>'^p = 1. c c a 196 QUATERNIONS, [chap. VIII. 32. Tangent cones are drawn from every point of 5(p— a)^(p— o) = «S to the similar and similarly situated surface Sp<l>p = 1, show that their planes of contact envelop the surface (5o<^p— !)• =in^8f4p. m 33. Find the envelop of planes which touch the parabolas ps=at*+pt, p^zar^-^-yr, where a, p, y form a rectangtdar system^ and t and r are scalais. 34. Find the equation of the surface on which lie the lines of contact of tangent cones drawn from a fixed point to a series of similar^ similarly situated^ and concentric ellipsoids. 35. Discuss the surfaces whose equations are SapSpp = Syp, and S^ap-k-S.app = 1. 36. Show that the locus of the vertices of the right cones which touch an ellipsoid is a hyperbola. CHAPTER IX. GEOMETRY OF CURVES AND SURFACES. 279. We have already seen (§ 31 (l)) that the equations p = </)^=2.a/(0, and p = <l>{t, u) = ^.af{ti u), where a represents one of a set of given vectors, and/ a scalar function of scalars t and u, represent respectively a curve and a surface. We commence the present brief Chapter with a few of the immediate deductions from these forms of expression. We shall then give a number of examples, with little attempt at systematic development or even arrangement. 280. What may be denoted by t and u in these equations is, of course, quite immaterial : but in the case of curves, con- sidered geometrically, t is most conveniently taken as the length, s, of the curve, measured from some fixed point. In the Kine- matical investigations of the next Chapter t may, with great convenience, be employed to denote time. 28L Thus we may write the equation of any curve in space as where <^ is a vector ftmction of the length, *, of the curve. Of course it is only a linear frmction when the equation (as in § 31 (A)) represents a straight line. 282. We have also seen (§§ 38, 39) that 198 QUATERNIONS. [CHAP. II. is a vector of unit length in the direction of the tangent at the extremity of p. At the proximate point, denoted hy s+is, this onit tangent vector becomes ^'*-|.<^''*d*+&c. But, because y^'^ - 1^ we have S^if/s^t'^s sz O, Hence (f/^s is a vector in the oscukting phme of the curve, and perp^idicukr to the tangent. Also, if M be the angle between the successive tangents 4>* and ip^^-^if/'ibS'^ , we have •cl = ^*"'> so that tie temar of ^'s is the reciprocal of tie radius ofalsohU curvature at tie point s. 288. Thus, if OP =r ^« be the vector of any point P of the curve, and if C be the centre of curvature at P, we have and thus OC = ^« 77- <!> s is the equation of the locus of the centre of curvature. Hence also Kif/sif/'s or <^'#</)"# is the vector perpendicular to the osculating plane ; and is the tortuosity of the given curve, or the rate of rotation of its osculating plane per unit of length. 284. As an example of the use of these expressions let us Jind tie curve wiose curvature and tortuosity are both constant. We have curvature = T^''s :sz Tp'z=i c. SECT. 284.] GEOMETRY OP CURYES AND SURFACES. 169 Hence <^'«<^''* = //p"= co, where a is a unit vector perpendicular to the osculating plane. This gives if c^ represent the tortuosity. Integrating we get p'ff'^Cp'^p, (1) where j8 is a constant vector. Squaring both sides of this equation, we get or T^ = \/c«+<?;. Multiply (1) by p', remembering that Tp'= 1, and we obtain ^p'z=, —Ci +p'j3, or p'= c^s—pP + a, (2) where a is a constant quaternion. Eliminating p\ we have of which the vector part is p"-.p/3» = -c^sp'^ Fap. The complete integral of this equation is evidently p = fcos,*y^+t;sin.*y^-^(e?»*/3-hrai3), (3) ^ and rj being any two constant vectors. We have also by (2), Spp = CiS-{-8a, which requires that Spi = 0, S^ri = 0. The farther test, that ^'= 1, gives us - 1 = r^«(f»sin».«IS3 + t;»cos».«T^- 2-S^sin.*T^cos.*2)3) ^ c> + ej 200 QUATERNIONS. [cHAP, IX, This requires^ of course^ ^-«> ^=^=3^ so that (3) becomes the general equation of a helix traced on a right cylinder. (Compare § 31 {m).) 285. The vector perpendicular from the origin on the tan- gent to the curve p = 4>8 is, of course, ^ f' ^, /r^ / -rfpp, or pFpp P (since p' is"a unit vector). 7b find a common property of curves whose tangents are all equidistant from the origin. Here TVpp = c, which may be written ^p^^S*pp'=c* (1) This equation shows that, as is otherwise evident, every curve on a sphere whose centre is the origin satisfies the condition. For obviously — p* = c" gives 8pp'z=, 0, and these satisfy (1). If 8pp' does not vanish, the integral of (1) is ^Tp^-c^ = s, (2) an arbitrary constant not being necessary, as we may measore s from any point of the curve. The equation of an involute which commences at this assumed point is -BJ = p — *p'. This gives JW« = Tp^ + «« - 2s8pp' = Tp^+s^^2s^/T^^^^\ by (1), = c\ by (2). This includes all curves whose involutes lie on a sphere about tie origin. 8ECT, 287.] GEOMETRY OF CUfiVES AND SURFACES. 201 286. Mnd the locus of tie foot of the perpendicular drawn to a tangent to a right helix from a point in the axis. The equation of the helix is s ^ . s p = acos- H-jSsin- -\-ys, a a ' where the vectors a, )3, y are at right angles to each other^ and Ta = 2)3 = h, while aTy = ^a'^-h^. The equation of the required locus is^ by last section, isr = pTp/ / s tf«— i« . *v ^/ . « a»— J* s^ . *• = a(cos-H ;— *sin-)+i3(sm — *cos-)+ y-r*. ^ a a" a^ ^^ a a* a/ ' a^ This curve lies on the hyperboloid whose equation is S^am^-S^^—a^S^ym = **, as the reader may easily prove for himself. 287. To find the least distance between consecutive tangents to a tortuous curve. Let one tangent be tsr = p-\-Xffy then a consecutive one, at a distance Is along the curve, is The magnitude of the least distance between these lines is, by §§ 203, 210, Trp'p"h» ' •if we neglect terms of higher orders. Dd 202 QUATERNIONS. [CHAP. IX. It may be written^ since pp" is a vector, and 7/= 1, ^s.vp''rpr\ But ^^^^y - r''^>'";u - ^ n'^.vv- Hence ^^s.Up^^FpY UVpY Vpr bs_ Tp'' is the small angle, d^, between the two successive positions of the osculating plane. [See also § 283.] Thus the shortest distance between two consecutive tangents is expressed by the formula 12r ' where r, = ^tttj is the radius of absolute curvature of the tor- tuous curve. 288. Let us recur for a moment to the equation of the parabola (§ 31 (/)) p^at^^. Here p'= {a+^f)-^. whence, if we assume Sa^ = 0, d» I T- = V— o*— /3'^», firom which the length of the arc of the curve can be derived in terms of t by integration. Again, p"= («+^0 -^ +fi{-^)'' But ^-± 1 _ , ^ S.p{a+^ d»* ~ ds' T{a+pi) ~ ^ ds T{a+^()' ' SECT. 289.] GEOMETRY OF CURVES AND SURFACES. 203 and therefore, for the vector of the centre of cterrature we have (§ 283), which is the quaternion equation of the evolute. 289. One of the simplest forms of the equation of a tortuous curve is where a, )8, y are any three non-coplanar vectors, and the numerical factors are introduced for convenience. This curve lies on a parabolic cylinder whose generating lines are parallel to y; and also on cylinders whose bases are a cubical and a semi-cubical parabola, their generating lines being parallel to j3 and a respectively. We have by the equation of the curve yP V dt ,'=(.+,<+ i-)^. from which, by Tp=^ 1, the length of the curve can be found in terms of ^j and from which p" can be expressed in terms of *. The investiga- tion of various properties of this cuirve is very easy, and will be of great use to the student. It is to be observed that in this equation t cannot sfemd for *, the length of the curve. Such an equation as or even the simpler form p = cu?+^*% involves an absurdity. 1} da 204 QUATEBKION& {chap. H. 290. The equation _ ^i^ where ^ is a self-conjugate linear and vector fonetion^ t sl scalar yariable, and c an arbitrary vector constant^ denotes a curious class of curves. We have at once where \og4> is another self*conjugate linear and vector fiinction, which we may denote by x* These functions are obviously commutative^ as they have the same principal set of rectangular vectors, hence we may write dp which of course gives As a verification, we should have = (i+8^x + Ux*+ )p where e is the base of Napier's Logarithms. This is obviously true if <^« = e^x, or = ex, or log<^ = x, which is our assumption. [The above process is, at first sight, rather startling, but the student may easily verify it by writing, in accordance with the results of Chapter V, whence <f>U = — /la/Sae— ^,0/S/3€— ^iyiSye. SECT. 293.] GEOMETRY OP CURVES AiTD SURFACES. 805 He will find at once and the results just given follow immediately.] 29L That the equation represents a surface is obvious from the fact that it becomes the equation of a definite curve whenever eitAer t or u has a particular value assigned to it. Hence the equation at once fiimishes us with two systems of curves, lying wholly on the surface, and such that one of each system can, in general, be drawn through any assigned point on the surface. Tangents drawn to these curves at their point of intersection must, of course, lie in the tangent plane, whose equation we have thus the means of forming. 292. By the equation we have where the brackets are inserted to show that these quantities are partial differential coefficients. If we write this as dp = <l>t dl + <f>u du^ the normal to the tangent plane is evidently and the equation of that plane 293. As a simple example, suppose a straight line to move along a fixed straight line, remaining always perpendicular to it, while rotating about it through an angle proportional to the space it has advanced; the equation of the ruled surface de- scribed will evidently be /) = o/ + wOcos^-f ysin^), (1) 206 QUATERNIONS. [cHAP. IX. where a, fi,y Bie rectangular vectors, and Tp=Ty. This surface evidently intersects the right cylinder p = a(j8cos^+y8in^)-f ra, in a helix (§§ 31 {m), 284) whose equation is p = o^-|-a(j8cos^-|-ysin^). These equations illustrate very well the remarks made in §§ 31 {l)f 291 J as to the curves or sur&ces represented by a vector equation according as it contains one or two scalar variables. Prom (1) we have dp = [o— «()8sin^— ycos^)]rf^+(/3cos^-fysin^rf«, so that the normal at the extremity of p is To (y cos ^— /3 sin ^)-«Jj3» J/o. Hence, as we proceed along a generating line of the surface, for which t is constant, we see that the direction of the normal changes. This, of course, proves that the surface is not de- velopable. 284. Hence the criterion for a developable surface is that if it be expressed by an equation of the form p = 4>t + uylrt, where ^ and yjrt are vector functions, we must have the direction of the normal F{<t/t^uyl/t}ylft independent of u. This requires either Fyjrt^^t =zO, which would reduce the surface to a cylinder, all the generating lines being parallel to each other ; or r<l/tylrtz=z 0. This is the criterion we seek, and it shows that we may write, for a developable surface in general, the equation p = 0^-f«^7 (1) SECT. 296.] GEOMETRY OF CURVES AND SURFACES. 207 Evidently p =z <^t is a curve (generally tortuous) and <^'^ is a tangent vector. Hence a developable surface is the locus of all tangent lines to a tortuous curve. Of course the tangent plane to the surface is the osculating plane at the corresponding point of the curve; and this is indicated by the fact that the normal to (1) is parallel to Vi^'t^"t (See § 283.) 295. A Geodetic line is a curve drawn on a surface so that its osculating plane at any point contains the normal to the surface. Hence, if i; be the normal at the extremity of p, p and p" the first and second differentials of the vector of the geodetic, which may be easily transformed into r.vdUp'^ 0. 296« In the sphere JJ) = a we have hence S.ppp''^ 0, which shows of course that p is confined to a plane passing through the origin, the centre of the sphere. For a formal proof, we may proceed as follows — The above equation is equivalent to the three Sep = 0, S0p = 0, S0p'-O, from which we see at once that ^ is a constant vector, and therefore the first expression, which includes the others, is the complete integral. Or we may proceed thus — = -p5.ppV'+/>"&pV= r.rppTpp''= F.Fpp'dFpp', whence by § 133 (2) we have at once UFpp^=z const. = suppose, which gives the same results as before. 208 QUATBKNIONS. [chap. IX. 287* Ib aDj cone we have^ of course^ 8vp = 0, since p lies in the tangent plane. But we have also 8vp = 0. Hence^ by the general equation of § 295^ eliminating v we get = (hS.pp'Fpy= SpdUp by § 133 (2). Integrating C^SpUp'-^jsdpUp'^z SpUp' -^JTdp. The interpretation of this is^ that the length of any arc of the geodetic is equal to the projection of the side of the cone (drawn to its extremity) upon the tangent to the geodetic. In other wordsj when the cane is developed the geodetic becomes a straight line. A similar result may easily be obtained for the geodetic lines on any developable surface whatever. 298. To find the shortest line connecting two points on a given surface. Here / Tdp is to be a minimum^ subject to the condition that dp lies in the given surface. Now hJTdp = jhTdp = -/^^^ = -fs.Udpddp = -[5. Udp 5p] +J8.bp d Udp, where the term in brackets vanishes at the limits^ as the extreme points are fixed^ and therefore bp = 0. Hence our only conditions are /' ' S.bpdUdp = 0, and Svbp = 0, giving V.vdUdp = 0, as in § 295. If the extremities of the curve are not given, but are to lie SHOT. 300.] GEOMETRY OF CURVES AND SURFACES. 200 on given curves, we most refer to the integrated portion of the expression for the variation of the length of the arc. And ^*s^<>™ S.Udphp shows that the shortest line cuts each of the given curves at right angles. 299. The osculating plane of the curve is 8.<f/t<l/'t{^'-p) = 0, (1) and is^ of course^ the tangent plane to the surface p = (l>t+u(l/t : (2) Let us attempt the converse of the process we have, so far, pursued, and endeavour to find (2) as the envelop of the variable plane (1). Differentiating (1) with respect to t only, we have &0V(tir-p) = 0. By this equation, combined with (1), we have or -orsp+iw^'sr 0+«^^'> which is equation (2). 300. This leads us to the consideration of envelops generally, and the process just employed may easily be extended to the problem (yi finding the envelop of a series of surfaces whose equa- tion contains one scalar parameter. When the given equation is a scalar one, the process of finding the envelop is precisely the same as that employed in ordinary Cartesian geometry, though the work is often shorter and simpler. If the equation be given in the form p = yjf{t, «, r), E e 210 QUATBRKIONS. [cttA]». IX. where ^ is a Teetor fimctioili i and « the scalar Yariables for any one sorfiu^^ 9 the soaku* parameter^ we have for a proximate surface Hence at all points on the intersection of two successiye sm&ces of the series we have which is equivalent to the following scalar equation connecting the quantities t, n, and v ; This equation, along with enables us to eliminate t, u, p, and the resulting scalar equation is that of the required envelop. SOL As an example, let us find the envelop of the osculating plane of a tortuous curve. Here the equation of the plane is (§299), 5.(tir-p)<^7<^''^ = 0, or «r = 4>t'^x<l/t'^y4rt = ^(ir,y, f), if p zs (t>t be the equation of the curve. Our condition is, by last seotion, or S4't 4/'t lift'\^xi^''t+y^'''f\ = 0, or y84t^"i^"H^ 0. Now the scalar factor cannot vanish, unless the given curve be plane, so that y = and the envelop is the developable surface, of which the given curve is the edge of regression, as in § 299. flBCT. 308.] GEOMETRY OP CITRVBS AND SUEFACBS. 211 302. When the equation contains two scalar parameters its differential coefficients with respect to them must vanish^ and we have thus three equations from which to eliminate two numerical quantities. A very common form in which these two parameters appear in quaternions is that of an unknown unit- vector. In this case the problem may be thus stated — Find the envelop ofihe mrface whose scalar expiation is where a is subject to the one condition Ta s 1. JDifferentiating with respect to a alone^ we have Svda = 0, Soda = 0, where r is a known vector frmction of p and a. Since da may have any of an infinite number of values^ these equations show *^* Vav = 0. This is equivalent to two scalar conditions only, and these, in addition to the two given scalar equations, enable \ui to eliminate a. With the brief explanation we have given, and the examples which follow, the student will easily see how to deal with any other set of data he may meet with in a question of envelops. 303. Find the envelop of a plane whose distance from the origin is constant Here Sap^-^c, with the condition Ta =z I. Hence, by last section, Fpa = 0, and therefore p = ^a, or Tp = c, the sphere of radius c, as was to be expected. 212 QUATEENIONS. [chap. IX. If we seek the envelop of thoee only of the planei which are parallel to a given vector fi, we have the additional relation 8afi:=z0. In this case the three differentiated equations are Spda = 0, Sada = 0, Spda = 0, and they give S.afip = 0. Hence a=:U.pFpp, and the envelop is TFpp = cTp, the circular cylinder of radius c and axis coinciding with ^. By putting Safi = e, where ^ is a constant different from zero^ we pick out all the planes of the series which have a definite inclination to fi, and of course get as their envelop a right cone. 304. The equation 8*ap + 28.app = * represents a parabolic cylinder, whose generating lines are parallel to the vector aFafi, For it is not altered by increasing p by^he vector xaFafi; also it cuts planes perpendicular to a in one line, and planes perpendicular to Fafi in two parallel lines. Its form and position of course depend upon the values of a, A and b. It is required to find its envelop if fi and b be constant, and a be subject to the one scalar condition Ta= 1. The process of § 302 gives, by inspection, p/Sap+ Ffip = xa. Operating by &a, we get S^ap-^-S.aPp =— ar, which gives S.afip = ar + i. But, by operating successively by S, Ffip and by &/), we have {FfipY = xSmPp, and {p^—x)Sap = 0, SECT. 306.] GEOMETRY OF CURVES AND SURFACES. 213 Omitting^ for the present^ the factor Sap, these three equations give, by elimination of a? and a, which is the equation of the envelop required. This is evidently a surface of revolution of the fourth order whose axis is fi ; but, to get a clearer idea of its nature, put <?» p-* = tsr, and the equation becomes which is obviously a surface of revolution of the second degree, referred to its centre. Hence the required envelop- is the reciprocal of such a surface, in the sense that the rectcmgle under the lengths of condirectional radii of the two is constant We have a curious particular case if the constants are so related that j+iS* = for then the envelop breaks up into the two equal spheres, touching each other at the origin, p« = ± Sfip, while the corresponding surface of the second order becomes the two parallel planes 305. The particular solution above met with, viz. 8ap = 0, limits the original problem, which now becomes one of finrling the envelop of a line instead of a surface. In fact this equation, taken in conjunction with that of the parabolic cylinder, belongs to that generating line of the cylinder which is the locus of the vertices of the principal parabolic sections. Our equations become Sap = 0, 214 QITATEBKIONS. [CHAP. IX. 2b =1; whence Vfip = a?o, giving X ^ '■^8,ctpp ^ — — > and thenoe * TFfip = -; so that the envelop is a circular cylinder whose axis is ^. [It is to be remarked that the equations above require that Safi = 0, so that the problem now solved is merely that of the envelop of a parabolic cylinder wAicA rotates about its focal line. This dis- cussion has only been entered into for the sake of explaining a peculiarity in a former result, because of course the present results can be obtained immediately by an exceedingly simple process.] 806« The equation 8ap8.afip = a^, with the condition Ta = 1, represents a series of hyperbolic cylinders. It is required to foul tAeir envelop. As before, we have p8.app 4- Vpp 8ap = xa, which by operating by S.a, 8.pf and 8.Fpp, gives 2a* =— ar, p*8.afip =s x8apj {Fppy8ap szx8.afip. Eliminating a and x we have, as the equation of the enve p"(r/3p)« = 4 a*. Comparing this with the equations p»=-2aS and (r/3p)«=-2aS SWJT. 307.] GEOMETRY OF CURVES AND SURFACES. 215 ivfaich represent a sphere and one of its circumscribing cylinders^ we see that^ if condirectional radii of the three surfaces be drawn from the origin, that of the new surface is a geometric mean between those of the two others. 307. -K^^ the envelop of all spheres which touch one given line and have their centres in another. Let p = P+yy be the line touched by all the spheres, and let xa be the vector of the centre of any one of them, the equation is (by § 200, or § 201) y^ip^xay = --(r.y (i3-^a))% or^ putting for simplicity, but without loss of generality, IV = 1, Safi=:0, Sfiy = 0, so that p is the least vector distance between the given lines, and, finally, p*— j3*-r2a?51ap = x^S^ay, Hence, by § 300, — 2/Sap = 2x8*ay, [This gives no definite envelop if Say = 0, i. e. if the line of centres is perpendicular to the line touched by all the spheres.] Eliminating x, we have for the equation of the envelop iS»ap + ^"ay(p»-/3») = 0, which denotes a surface of revolution of the second degree, whose axis is a. Since, from the form of the equation, Tp may have any magnitude not less than T/3, and since the section by the plane Sap=0 is a real circle, on the sphere the surface is a hyperboloid of one sheet. 216 QUATBBKIOKS. [chap. IX. [It will be instructive to the student to find the signs of the values of ^1,^,,^, as in § 165^ and thence to prove the above conclusion.] 308. As a final example let us find the envelop of the hyper- bolic cylinder Sap Sfip^c = 0, where the vectors a and /3 are subject to the conditions Ta = 253 = 1, Say = 0^ Sfib = 0. [It will be easily seen that two of the six scalars involved in a, p still remain as variable parameters.] We have Soda = 0, Syda = 0, so that da = xFay. \ Similarly dp =s yFfib. But^ by the equation of the cylinders^ Sap Spdfi + Spda Sfip = 0, or ySap S.pbp -\-x8.ayp Spp = 0. Now by the nature of the given equation^ neither Sap nor Sfip can vanish^ so that the independence of da and dp requires . . S.ayp = 0, S.pbp = 0. Hence a = U.yFyp, p =iU,h7hp, i and the envelop is I T.Vypnp-cTyb ^ 0, a surface of the fourth order, which may be constructed by laying off mean proportionals between the lengths of condirec- tional radii of two equal right cylinders whose axes meet in the | ongm. I I 309* We may now easily see the truth of the following general statement. SECT. 310.] GEOMETRY OF CURVES AND SURFACES. 217 Suppose the given equation of th^ series of surfaces, whose envelop is required, to contain m vector, and n scalar, para- meters ; and that the latter are subject to j» vector, and j scalar, conditions. In all there are Bm + n scalar parameters, subject to 3p-\-q scalar conditions. That there may be an envelop we must therefore in general *^av^ {Sm+n)-^{3p + q)=zl, or =2. In the former case the enveloping surface is given as the locus of a series of curves, in the latter a series ot points. Differentiation of the equations gives us 3jo-f j-f 1 equations, linear and homogeneous in the 3m-\-n differentials of the scalar parameters, so that by the elimination of these we have one final scalar equation in the first case, two in the second; and thus in each case we have just equations enough to eliminate all the arbitrary parameters. 310. To find the locus of the foot of the perpendicular drawn from the origin to a tangent plane to any surface. If Svdp = be the differentiated equation of the surface, the equation of the tangent plane is 8{iff-^p)v = 0. We may introduce the condition 8vp = 1, which in general alters the tensor of v, so that v~^ becomes the required vector perpendicular, as it satisfies the equation Siffv = 1. It remains that we eliminate p between the equation of the given surface, and the vector equation tff = v~^. The result is the scalar equation (in w) required. Ff ^ 218 QUATBRNIONS. [chap. IX. For exsmplej if the given surface be the ellipsoid SfHt>p = 1, we have «•""* = y = ^, so that the required equation is iS«r-»^-*«r-» = 1, or Sw^'^^w = «r*, which is Fresnel's Surface of Elasticity. (§ 263.) 31L To find the reciprocal of a given surface with respect to the unit sphere whose centre is the origin. With the condition Spv = 1, of last section^ we see that — y is the vector of the pole of the tangent plane 8{w-p)v = 0. Hence we must put w =—v, and eliminate p by the help of the equation of the given surface. Take the ellipsoid of last section^ and we have w =— <^p, so that the reciprocal surfiEice is represented by 5W^~^«j = 1. It is obvious that the former ellipsoid can be reproduced £rom this by a second application of the process. And the property is general^ for Spv = 1 givesj by differentiation^ and attention to the condition Svdp = 0, the new relation Spdv = 0^ so that p and v are corresponding vectors of the two surfaces : either being that of the pole of a tangent plane drawn at the extremity of the other. SECT. 314.] GEOMETRY OF CURVES AND SURFACES. 219 312. If the given surface be a cone with its vertex at the origin, we have a peculiar case. For here every tangent plane passes through the origin, and therefore the required locus is wholly at an infinite distance. The difficulty consists in Spv becoming in this case a numerical multiple of the quantity which is equated to zero in the equation of the cone, so thai of course we cannot put as above Spv = 1. 313* The properties of the normal vector v enable us to write the partial differential equations of families of surfaces in a very simple form. Thus the distinguishing property of Cylinders is that all their generating lines are parallel. Hence all positions of v must be parallel to a given plane^or Sou = 0, which is the quaternion form of the well-known equation dF dF ^_o dx dy dz "^ ' To integrate it, remember that we have always 8vdp = 0, and that as i^ is perpendicular to a it may be expressed in terms of any two vectors, /3 and y, each perpendicular to a. Hence v = xfi+yy, and xSpdp + ySydp = 0. This shows that Sfip and Syp are together constant or together variable, so that Spp =/(8yp), where/ is any scalar function whatever. 314. In Surfaces of Revolution the normal intersects the axis. Hence, taking the origin in the axis a, we have 220 QUATERNIONS. [cHAP. IX. S.app = 0, or ' V = xa+yp. Hence xSadp+ySpdp = 0, whence the integral Tp =/{Sap). The more common form^ which is easily derived from that just written, is jfjr^^ ^ F{8ap). In Cone9 we have Svp = 0, and therefore Svdp = S.v{TpdUp+UpdTp) = TpSvdUp. Hence SvdUp = 0, so that V must be a function of Up^ and therefore the integral is nup) = 0, which simply expresses the fact that the equation does not involve the tensor of p. 315. y equal lengths be laid off on the normals drawn to any surface , the new surface formed by their extremities is normal to the same lines. For we have m = p+aUp, and Svdw^ = Svdp+aSvdUv = 0, which proves the proposition. Take, for example, the surface Spfpp = 1 ; the above equation becomes ad>p so that p = (^ -f l) 'fir, and the equation of the new surface is to be found by eliminating a -pjT— (written x) between the equations l(t>p SECT. 317.] GEOMETRY OP CURVES AND " SURFACES. 221 and = 8.<f>{af<t>+ l)-*tEr<^(a?<^+ l)-*tEr. 316. It appears from last section that if one orthogonal surface can be drawn cutting a given system of straight lines, an indefinitely great number may be drawn: and that the portions of these lines intercepted between any two selected surfaces of the series are all equal. Let p ^ (T-f a?r, where <t and r are vector fiinctions of p, and x is any scalar, be the general equation of a system of lines : we have Srdp = = S(p—(r)dp as the dijBTerentiated equation of the series of orthogonal surfaces, if it exist. Hence the following problem. 317. It is required to find the criterion of integraUlity of the equation g^^^ ^ ^ (^^ as the complete differential of the equation of a series of surfaces. Hamilton has given {Elements^ p. 702) an extremely elegant solution of this problem, by means of the properties of linear and vector fiinctions. We adopt a different and less simple process, on account of some results it offers which will be useful to us in the next Chapter ; and also because it will show the student the connection of our methods with those of ordinary differential equations. If we assume Fp ^ C to be the integral, and apply to it the very singular operator devised by Hamilton, _ , d , d ^ , d ,dF ,dF ^dF we have VFp =^ ^ t- +Jj- + ^'r ' 222 QUATERNIONS. [CHAP. IX. But p = iaf+J3f'\-kz, whence dp = idx-^jdy+kdz, and =dF= ^^+ ^^^^-^Tz^^ ^-^SVFdp. Comparing with the given equation^ we see that the latter represents a series of surfaces if v^ or a scalar multiple of it^ can be expressed as V^. If v = V^, we have V. = V.i' = -(^ + ^ + ^), a well-known and most important expression^ to which we shall return in next Chapter. Meanwhile we need only remark that the last-written quantities are necessarily scalars, so that the only requisite condition of the integrability of (1) is rVy=0 (2) K V do not satisfy this criterion^ it may when multiplied by a scalar. Hence the farther condition rV{wp) = 0, which may be written FvVw+wFVv = (3) This requires that SpVv s (4) If then (2) be not satisfied, we must try (4). If (4) be satisfied w will be found from (3); and in either case (1) is at onoe integrable. p[f we put dv = (t>dp where ^ is a linear and vector fiinction, not necessarily self- conjugatCj we have by § 173. Thus, if <^ be self-conjugate, c = 0, and the criterion 8B0T. 319.] GEOMETRY OF CURVES AND SURFACES. 223 (2) is satisfied. If ^ be not self-conjugate we have by (4) for the criterion g^^ _ These results accord with Hamilton's, lately referred to, but the mode of 6btaiQing them is quite different from his.] 318. As a simple example let us first take lines diverging from a point. Here r||p, and we see that ifv=^p so that (2) is satisfied. And the equation is Spdp = 0, whose integral Tp = C gives a series of concentric spheres. Lines perpendicular to, and mtersecling, a fixed line. If a be the fixed line, /3 any of the others, we have 8.c^p = 0, Safi = 0, 8^dp = 0. Here v || a Fop, and therefore equal to it, because (2) is satisfied. Hence 8.dpaVap = 0, or a*8pdp^Sap8adp = 0, whose integral is the equation of a series of right cylinders 319. To find tie orthogonal trajectories of a series of circles whose centres are in, and their planes perpendicular to, a given line. Let a be a unit- vector in the direction of the line, then one of the circles haa the equations Tp^C, ^ 8ap= Cr,S where C and C are any constant scalars whatever. • 224 QUATERNIONS. [chap. IX. Henoe^ for the required surfaces V II d^p 11 Vapy where d^p is an element of one of the circles, v the normal to the orthogonal surface. Now let dp be an element of a tangent to the orthogonal surface^ and we have Svdp = S.apdp =• 0. This showtf that dp is in the same plane as a and p, i. e. that the orthogonal surfaces are planes passing through the com- mon axis. [To integrate the equation S,apdp = evidently requires, by § 317, the introduction of a fiictor. For rVFap = F{irai+jraJ+irai) = 2a, so that the first criterion is not satisfied. But S.FapF^rap = 28.aFap = 0, so that the second criterion holds. It gives, by (3) of § 317, F.^wFap-\'2wa = 0, or pSaVw—aSp^w + 2wa = 0. That is SaVto = 0, ^ SpVw = 2w. J These equations are satisfied by But a simpler mode of integration is easily seen. Our equation may be written S.apdp = 0, 0=.S.aF^ = Sa^-^^dS.alogU^^ p Up ^ p which is immediately integrable, ^ being an arbitrary but con- stant vector. SECT. 32Q.] GEOMETRY OP CgEVBS AND SURFACES. 225 As we have not introduced into this work the logarithms of versors, nor the corresponding angles of quaternions^ we must refer to Hamilton's Elements for a farther development of this point.] 320. To find the orthogonal trajectories of a given series of surfaces. If the equation i^P = C, give &vdp = 0, the equationjf of the orthogonal curves is Yvdp = 0. This is equivalent to two scalar differential equations (§ 197)^ which^ when the problem is possible^ belong to surfaces on each of which the required lines lie. The finding of the requisite criterion we leave to the student. Let the surfaces he concentric spheres. Here p» = G, and therefore Ypd^ = 0. Hence dJJp = — UpVpdp = 0, and the integral is Up = constant^ denoting straight lines through the origin. Let the surfaces he spheres touching each other at a common paint. The equation is (§ 218) Sap-^ = C, whence V.papdp = 0. The integrals may be written S.afip = 0, p^ + hTFap = 0, the first (fi being any vector) is a plane through the common diameter; the second represents a series of rings or tores (§ 322) 226 QUATBRNIOK& [cwitf . IX formed by the revolation^ about a, of circles icnehing that line at the point common to the spheres. Let the surfaces be similar, similarly siiuated, and eonceniriej surfaces of tie second order. Here Spxp = C, therefore ^XP^P == ^' But^ by § 290; the integral of this eqnation » p = ^xe where ^ and x &i^ related to each other^ as in § 290 ; and e is any constant vector. 32L Find tie general equation of surfaces described by a line which always meets, at right angles j a fixed line, K a be the fixed line^ /3 and y forming with it a rectangular unit sjrstem^ then p = ara+jr(/3 + ^y), where y may have all values, but x and z are mutuaUy de- pendent, is one form of the equation. Another, expressing the arbitrary relation between x and z, is But we may also write as it obviously expresses the same conditions. The simplest case is when F{x) =s kx. The surface is one which cuts, in a right helix, every cylinder which has a for its axis. 322. The centre of a sphere moves in a given circle, fiid the equation of the ring described. Let a be the unit-vector axis of the circle, its centre the origin, r its radius, a that of the sphere. SKCT. 323.] GEOMETRY OF CURVES AND SURFACES. 227 Then (p-fiY - -«^ is the equation of the sphere in any position, where «a^ = 0, Tfi = r. These give S.afip = 0, and fi must now be eliminated. The result is that ^ - raUVap, giving (p« — /-a + a^'Y z= 4:r''T^ Fap, which is the required equation. It may easily be changed to (p«— ^a + r^)' =:-4eiap»— 4r»iS'»ap, (1) and in this form it enables us to give an immediate proof of the very singular property of the ring (or tore) discovered by ViUarceau. For the planes S.p(a±— !i_) = 0, which together are represented by f^{r^^a^)8'ap^a^8^pp = 0, evidently pass through the origin and touch (and cut) the ring. The latter equation may be written r»S»ap-a^{8^ap + S''pUfi) = 0, or r^S^ap+a^ip^-^-S^apUp) :=: (2) The plane intersections of (1) and (2) lie obviously on the new surface (p^^a^ + f^Y = ^a^S^.apUp, which consists of two sphwes of radius r, as we see by writing its separate factors in the form (p±aaUpy + r'' = 0, 323. It may be instructive to work out this problem from a diff^^nt point of view, especially as it affords excellent prac- tice in transformations. G g ^ 228 QUATERNIONS. {CHAP. li A circle revolver about- an tuns passing within it, tie perpen- dicular from the centre on the axis lyifig in the plane of the circle : show that, for a certain position of the axis, the same solid may be traced out by a circle revolving about an external axis in its own plane. Let a = ^/4^+? be the radius of the circle, i the vector axis of rotation, — (?a (where Ta = 1) the vector perpendicidar from the centre on the axis i, and let the vector bi-\-cia be perpendicular to the plane of the circle. The equations of the circle are (p— tfa)* -h i« + c« = 0, 1 S(i + I ia) p = 0. J Also — p« = 8Hp + S^a(i -^-SKiap, b* = SHp^8^ap'{-—SHp by the second of the equations of the circle. But, by the first, which is easily transformed into {p^^b^y =z^Aa^{p^ + 8Hp), or p'-^b* zzz^2aTFip. If we put this in the forms p»-i« = 2a8pp, and (p— flf)3)» + c* = 0, where ^ is a unit- vector perpendicular to i and in the plane of i and p, we see at once that the surface will be traced out by a circle of radius c, revolving about i, an axis in its own plane, distant a from its centre. This problem is not well adapted to show the gain in brevity and distinctness which generally follows the use of quaternions ; SECT. 325.] GEOMETRY OF CURVES AND SURFACES. 229 as, from its very nature, it hints at the adoption of rectangular axes and scalar equations for its treatment, so that the solution we have given is but little different from an ordinary Cartesian one. 324. A surface is geiieraied ly a straight line which intersects two fixed Unes : fimd the genial equation. If the given lines intersect, there is no surface but the plane containing them. Let then their equations be. Hence every point of the surface satisfies the condition, § 30, 9 = 5^(a+a?i8)4-(l-J^)(a,+^ii3i) (1) Obviously y may have any value whatever: but to specify a psurticular surface we must have a relation between x and x^. By tbe help of this, x^ may be eliminated from (1), which then takes the usual form p^^{x,y\ Or we may operate on (1) by r'.(a+ir/3--ai— iFijSi), so that we get a vector equation equivalent to two scalar equations (§ 98), and not containing y. From this x and a?i may easily be found in terms of p, and the general equation of the possible surfeces may be written f{Xy X^ = 0, where/ is an arbitrary scalar function, and the values of x and Xy are expressed in terms of p. This process is obviously applicable if we have, instead of two straight lines, any two given curves through which the line must pass; and even when the tracing line is itself a given curve, situated in a given manner. But an example or two will make the whole process clear. 325. Suppose the moveable line to be restricted by the condition that it is always parallel to afi^edplane. 230 QUATERNIONS. [chjs:p. IXt Then, in addition to (1)^ we have the condition 5y(at+«ti3x-o-ar)8) = 0, y being a vector perpendicular to the fixed plane. We lose no generality by asBiiming a and a^, which are any vectors drawn from the origin to the fixed lines^ to be each perpendicular to y; for, if for instance we could not assume Sya = 0, it would follow that SyP = 0, and the required surfiace would either be impossible, or would be a plane, cases which we need not consider. Hence ^iSyfif^Syp = 0. Eliminating t^, by the help of this equation, from (1) of last section, we have Operating by any three non-coplanar vectors, we obtain three equations from which to eliminate x and y. Operating by S.y we find Syp = xSfiy. Eliminating x by means of this, we have finally which appears to be of the third order. It is really, however, only of the second order, since, consistently with our assump- tions, we have jr^^ y ^^ and therefore Syp is a spurious factor of the left-hand side. 326. IJet the fixed Ivaee be perpendicular to each other, and let the moveable line pass through the drcumferenee of a circle, whose centre ie in the common perpendicular, and whose plane bisects that line at right angles. Here the equations of the fixed lines may be written p = a-\-xp, p =— oH-a?i.y, SECT.^27,] GEOMETRY OF CITRVES AND SURFACES. 231 where a, fi, y form a rectangular system^ and we may assume the two latter to be unit- vectors. The circle has the equations p« =— «% Sap = 0. Equation (1) of § 324 becomes P=y(a + a?^) + (l— y)(-a-|-«?iy). Hence Sa-'^p = ^—(1 — ^) = 0, or ^ = h Also p= =— a» = (2y-l)«a» — ;ry-ar?(l— ^)S or 4 a' = (iJ?'-f-^i), ' so that if we now suppose the tensors of fi and y to be each 2 a, we may put a? = cos d, x^ = sin 6, from which p = (2y— l)aH-^/3cos^+(l— ^)ysin^; and finally -j- — P^^-rr + ,:. — o'^r^:, = 4a*. For this very simple case the solution is not better than the ordinary Cartesian one; but the student will easily see that we may by very slight changes adapt the above to data far less symmetrical than those from which we started. Suppose, for instance, p and y not to be at right angles to one another; and suppose the plane of the circle not to be parallel to their plane, &c., &c. But farther, operate on every line in space by the linear and vector function <^, and we distort the circle into an ellipse, the straight lines remaining straight. If we choose a form of <^ whose principal axes are parallel to a, p, y, the data wiU remain symmetrical, but not unless. This subject will be considered again in the next Chapter. 327. To find the curvature of a normal section of a central surface of the second order. In this, and the few similar investigations which foUow, it will be simpler to employ infinitesimals than difi*erentials ; though 232 QUATEBNIONS. [chap. IX- for a thorough treatment of the subject the latter method^ bb maj be seen in Hamilton's ElemenU, is preferable. We have, of coursej Spi^p = 1, and, if p+Sp be also a vector of the surface, we have rigorously, whatever be the tensor of bp, 5(P+»P)*(P+»P) = 1- Hence 2Sbp<l>p-{-Shpipbp = (1) Now ^p is normal to the tangent plane at the extremity of p, so that if t denote the distance of the point p + bp from that plane ^ ^ SbpU<l>p, and (1) may therefore be written 2tHl>p^T*bp8.Uip<l>mp = 0. But the curvature of the section is evidently or, by the last equation. In the limit, dp is a vector in the tangent plane ; let cr be the vector semidiameter of the surface which is parallel to it, and the equation of the sm&ce gives T^^S.U^U'SF = I, so that the curvature of the normal section, at the point p, in the direction of cr, is 1 directly as the perpendicular from the centre on the tangent plane, and inversely as the square of the semidiameter parallel to the tangent line, a well-known theorem. .328. By the help of the known properties of the central SECT. 328.] GEOMETRY OP CURVES AKD SURFACES. 233 section parallel to the tangent plane^ this theorem gives us all the ordinary properties of the directions of maximum and minimum curvature, their being at right angles to each other, the curva- ture in any normal section in terms of the chief curvatures and the inclination to their planes, &c., fee, without farther analysis. And when, in a future section, we show how to find an osculating surface of the second order at any point of a given surface, the same properties will be at once established for surfaces in general. Meanwhile we may prove another curious property of the surfaces of the second order, which similar reasoning extends to all surfaces. The equation of the normal at the point p+5p in the surface treated in last section is isr = p'\-lp + x<t>{p-\-hp) (1) This intersects the normal at p if (§§ 203, 210) S.bp(l>p(l)bp = 0, that is, by the result of § 273, if 6p be parallel to the maximum or minimum diameter of the central section parallel to the tangent plane. Let (Ti and o-, be those diameters, then we may write in &^^e^*al bp =pa;-^qir„ where jD and q are scalars, infinitely small. If we draw through a point-P in the normal at p a line parallel to o-i, we may write its equation -BT = p-\'(l<l>p+ya'i. The proximate normal (1) passes this line at a distance (see § 2^^) 8.{a4>p^hp)UV(T^^{p-^hp), or, neglecting terms of the second order, •■^p^—{ap8.<l>pa^<l>a^-]-aqS.(l)par^<t>ar^ + qS,a^(ri(l>p). The first term in the bracket vanishes because a^ is a principal Hh 234 QUATBRKIONS. [cHAP. IX. ▼ector of the section parallel to the tangent plane^ and thus the expression becomes Hence^ if we take a s T<r\y the distance of the normal from the new line is of the second order only. This makes the distance of P from the point of contact T^^p Ta\y i. e. the principal radius of curvature along the tangent line parallel to <r,. That is the group of normals drawn near a point of a central surface of the second order pass ultimately throttgh two lines each parallel to the tangent to mie principal section, and passing through the centre of curvature of the other. The student may form a notion of the nature of this proposition by considering a small square plate^ with normals drawn at every point, to be slightly bent, but by different amounts, in planes perpendicular to its edges. The first bending will make all the normals pass through the axis of the cylinder of which the plate now forms part ; the second bending will not sensibly disturb this arrangement, except by lengthening or shortening the line in which the normals meet, but it will make them meet also in the axis of the new cylinder, at right angles to the first. A small pencil of light, with its focal lines, presents a similar appearance. 329. To extend these theorems to surfaces in general, it is only necessary, as Hamilton ^has shown, to prove that if we ^*« dv = i>dp, <^ is a self-conjugate function ; and then the properties of <f>, as explained in preceding Chapters, are applicable to the question. As the reader will easily see, this is merely another form of the investigation contained in § 317. But it is given here to show what a number of very simple demonstrations may be given of almost all quaternion theorems. Now V is defined by an equation of the form dfp = Svdp, SECT. 331.] GEOMETRY OF CURVEd AND SURFACES. 235 where / is b, scalar fanction. Operating on this by another independent symbol of differentiation, b, we have bdfp = Sbvdp+Svhdp* In the same way we have db/p = Sdvbp+Svdbp, But, as d and b are independent, the left-hand members of these equations, as well as the second terms on the right (if these exist at all), are equal, so that we have Sdvbp = Sbvdp, This becomes, putting dv = Kpdp, and therefore bv = <t>bp, Sbp<f>dp = Sdp<l>bp, which proves the proposition. 330. If we write the differential of the equation of a surface in the form dfp^2Svdp, then it is easy to see that /{p + dp) =z/p-\-2Svdp + 8dvdp-\-&c., the remaining terms containing as factors the third and higher powers of Tdp, To the second order, then, we may write, except for certain singular points, = 2Svdp-\-8dvdp, and, as before, (§ 327), the curvature of the normal section whose tangent line is dp is Id, r,dp' SSL The step taken in last section, although a veiy simple one, virtually implies that the first three terms of the expan- sion o{ f{p-\-dp) are to be formed in accordance with Taylor^s Theorem, whose applicability to the expansion of scalar functions of quaternions has not been proved in this work, (see § 135); H h !% 236 QUATERNIONS. [cHAP. IX. we therefore give another investigation of the cu^rvature of a normal section^ employing for that purpose the formulae of § (282). We have^ treating dp as an element of a curve^ Svdp = 0, or^ making 9 the independent variable, 8vp' = 0. From this, by a second differentiation, S^p'+Si^p'' = 0. The curvature is, therefore, since v^p' and Tp = 1, 332. Since we have shown that dv = ffidp where ^ is a self-conjugate linear and vector function, whose constants depend only upon the nature of the surface, and the position of the point of contact of the tangent plane ; so long as we do not alter these we must consider ^ as possessing the properties explained in Chapter V. Hence, as the expression for Tp" does not involve the tensor of dpy we may put for dp bxlj unit- vector r, subject of course to the condition 8vT = (1) And the curvature of the normal section whose tangent is r is If we consider the central section of the surface of the second made by the plane Svzr = 0, we see at once that the curvature of ike gwen surface along the SECT. 334.] GEOMETRY OF CURVES AND SURFACES. 237 normal section touched by r is inversely as the square of the parallel radius in the auxiliary surface. This, of course, includes Euler^s and other well-known Theorems. 333. To find the directions of maximum and minimum curvature^ we have /Sr<^T = max. or min. with the conditions, 8VT = 0, ?V = 1. By differentiation, as in § 273, we obtain the farther equation 8.VTil>T = (1) If T be one of the two required directions rz=.rJJv is the other, for the last-written equation may be put in the form 8.tUv<I){vtUv) = 0, i.e. 8.T<l^(v/) = 0, or S,vT'(f)T' = 0. Hence the sections of greatest and hast curvature are perpendicular to one another. We easily obtain, as in § 273, the following equation 8.v{<l> + 8T(t>T)'^v = 0, whose roots divided by Tv are the required curvatures. 334. Before leaving this very brief introduction to a subject, an exhaustive treatment of which will be found in Hamilton's Elements y we may make a remark on equation (1) of last section 8.vt4)t = 0, or, as it may be written, by returning to the notation of § 332, S.vdpdv = 0. This is the general equation of lines of curvature. For, if we define a line of curvature on any surface as a line such that 238 QUATERNIONS. [chap. IX nonnals drawn at contigaous points in it intersect^ then, bp being an element of such a line, the normals fir = p-^xp and fir = p-^bp-\-y{v^bv) must intersect. This gives, by § 203, the condition S.bpvhi; = 0, as above. EXAMPLES TO CHAPTER IX. 1. Find the length of any arc of a curve drawn on a sphere so as to make a constant angle with a fixed diameter. 2. Show that, if the normal plane of a curve always contains a fixed line, the curve is a circle. 3. Find the radius of spherical curvature of the curve p = 4>t. Also find the equation of the locus of the centre of spherical curvature. 4. (Hamilton, Bishop Lwufa "Premium Examination^ 1854.) {a,) If p be the variable vector of a curve in space, and if the differential dK be treated as = 0, then the equation if^_^^ ^ o obliges ic to be the vector of some point in the normal plane to the curve. CHAP. IX.] EXAMPLES TO CHAPTER IX. 239 (4.) In like manner the system of two equations, where dK and d*K are each = 0, dT{p^K) = 0, J>y(p-ic) = 0, represents the axis of the element, or the right line drawn through the centre of the osculating circle, perpendicular to the osculating plane. (c.) The system of the three equations, in which k is ^ treated as constant, determines the vector k of the centre of the oscu- lating sphere. (d.) For the three last equations we may substitute the following; 8\p'-K)dp = 0, S.(p-^K)d^p-^3S.dpd'p = 0. (e.) Hence, generally, whatever the independent and scalar variable may be, on which the variable vector p of the curve depends, the vector k of the centre of the osculating sphere admits of being thus expressed : _ ^V.dpd^pS.dpd^p-dp^F.dpd^p ''-P-^ S.dpd^pd^p (/.) In general, did-^r.dpUp) = d(Tp-*r.pdp) = Tp-'{3F.pdp8.pdp^p'r.pd'p); whence, sr.pdpS.pdp-^p^F.pd'p = 'p^Tpdip-^r.dpUp); and, therefore, the recent expression for k admits of being thus transformed, _ dp*d{dp-^F. d^pUdp) "-P-^ S.d'pd'pUdp ' {ff,) If the length of the element of the curve be constant, dTdp = 0, this last expression for the vector of the 240 QUATBBNIONS. [chap. IX. centre of the osculating^ sphere to a curve of doahle curvature becomes^ more simply, , "" ^ ^'^ S.dpd^pd^p' or r-nJ. ^'^'P^P' ""^-^STd^dV (i.) Verify that this expression gives k = 0, for a curve described on a sphere which has its centre at the origin of vectors ; or show that whenever dTp = 0, d^Tp = 0, d^Tp = 0, as well as dTdp = 0, then pS.dp'^d^pd'p = V.dpd^p. 6. Find the curve from every point of which three given spheres will appear of equal magnitude. 6. Show that the locus of a point, the difference of whose distances from each two of three given points is constant, is a plane curve. 7. Find the equation of the curve which cuts at a given angle all the sides of a cone of the second order. Find the length of any arc in terms of the distances of its extremities from the vertex. 8. Why is the centre of spherical curvature, of a curve de- scribed on a sphere, not necessarily the centre of the sphere ? 9. Find the equation of the developable surface whose gene- rating lines are the intersections of successive normal planes to a given tortuous curve. 10. Find the length of an arc of a tortuous curve whose normal planes are equidistant from the origin. 11. The reciprocals of the perpendiculars from the origin on the tangent planes to a developable surface are vectors of a tortuous curve; from whose osculating planes the cusp-edge of the original surface may be reproduced by the same process. CHAP. IX.] EXAMPLES TO CHAPTER IX. 241 12. The equation where a is a unit- vector not perpendicular to j3, represents an ellipse. If we put y = Fa/3, show that the equations of the locus of the centre of curvatui*e are S.fiyp = 0, S^fip + S^yp = (fiSUafi)i. 13. Rnd the radius of absolute curvature of a spherical conic. 14. If a cone be cut in a circle by a plane perpendicular to a side^ the axis of the right cone which osculates it^ along that side, passes through the centre of the sectioiv. 15. Show how to find the vector of an umbilicus. Apply your method to the surfaces whose equations are aS/m^P = 1, and SapSppSyp = 1. 16. Find the locus of the umbilici of the surfaces represented by the equation 8p{<t> + Ay'p = 1, where ^ is an arbitrary parameter. 17. Show how to find the equation of a tangent plane which touches a surface along a line. Find such planes for the follow- ing surfaces Sp<l>p = 1, and (p^— a=»-f i»)» + 4(aV' + *'^'ap) = 0. 18. Find the condition that the equation S{p+a)<l>p=l, where is a self-conjugate linear and vector function, may represent a cone. I i 242 QUATEBNIONS. [CHAP. IX. 19. Show from the general equation that cones and cylinders are the only developable surfaces of the second order. 20. Find the equation of the envelop of planes drawn at each point of an ellipsoid perpendicular to the radius vector from the centre. 21. Find the equation of the envelop of spheres whose centres lie on a given sphere, and which pass through a given point. 22. Find the locus of the foot of the perpendicular from the centre to the tangent plane of a hyperboloid of one, or of two, sheets. 23. Hamilton, Bishop Lavfs Premium Examinationy 1852. (a.) If p be the vector of a curve in space, the length of the element of that curve is Tdp ; and the variation of the length of a finite arc of the curve is h/Tdp = -/iS TJdphdp = ^l^SUdphp +/SdUdphp. (b.) Hence, if the curve be a shortest line on a given sur- face, for which the normal vector is v, so that Svbp = 0, this shortest or geodetic curve must satisfy the differential equation, FpdUdp = 0. Also, for the extremities of the arc, we have the limiting equations, SUdp.bpo = ; SUdp^bpi = 0. Interpret these results. ((?.) For a spheric surface, Fvp = 0, pdUdp = ; the inte- grated equation of the geodetics is pUdp = w, giving Surp = (g^eat circle). For an arbitrary cylindric surface, 8av = 0, adUdp = ; the integral shows that the geodetic is generally a helix, making a constant angle with the generating lines of the cylinder. CHAP. IX.] EXAMPLES TO CHAPTER IX. 243 (d.) For an arbitrary conic surface, Svp = 0, SpdUdp = ; integrate this differential equation, so as to deduce from it, TFpUdp = const. Interpret this result; shew that the perpendicular from the vertex of the cone on the tangent to a given geodetic line is constant; this gives the rectilinear development. When the cone is of the second degree, the same property is a particular case of a theorem respecting confocal surfaces. {e,) For a surface of revolution, S.cLpv = 0, S.apdUdp = ; integration gives, const = S.apUdp = TFapSUiFapJp) ; the perpendicular distance of a point on a geodetic line from the axis of revolution varies inversely as the cosine of the angle under which the geodetic crosses a parallel (or circle) on the surface. (/.) The differential equation, S.apdUdp = 0, is satisfied not only by the geodetics, but also by the circles, on a surface of revolution ; give the explanation of this fact of calculation, and show that it arises from the coincidence between the normal plane to the circle and the plane of the meridian of the surface. (ff.) For any arbitrary surface, the equation of the geodetic may be thus transformed, S.vdpd^p — ; deduce this form, and show that it expresses the normal property of the osculating plane. (L) If the element of the geodetic be constant, dTdp = 0, then the general equation formerly assigned may be reduced to V.vd^p = 0. Under the same condition, d^p = —v-^Sdvdp. I i % 244 QUATERNIONS. [chap. IX. (>.) If the equation of a central surface of the second order be put under the form ^=: 1, where the fanetion/ is scalar, and homogeneous of the second dimension^ then the differential of that function is of the form dfy = 2S.vdp, where the normal vector, r = <^p, is a distributive function of p (homogeneous of the first dimension), dv = d<l>p = <f>dp. This normal vector v may be called the vector of prox^ imity (namely, of the element of the surface to the centre); because its reciprocal, r~*, represents in length and in direction the perpendicular let fall from the centre on the tangent plane to the sur&ce. (^.) If we make Sa^p z=.f{iTy p), thi% function f is com- mutative with respect to the two vectors on which it depends, y(p, o) =/(a', p) ; it is also connected with the former function f of a single vector p, by the relation, y(p, p) =/p : so thatyp = &p<^p. fdp = Sdpdv ; dfdp = 2S.dvd'p ; for a geodetic, with constant element, 2fdp V this equation is immediately integrable, and gives const. = Tv ^/{fUdp) = reciprocal of Joachimstal^s product, PL. (/.) If we give the name of " Didonia'' to the curve (dis- cussed by Delaunay) which, on a given surface and with a given perimeter, contains the greatest area, then for such a Didonian curve we have by quater- nions the formula, fS. Uvdphp -f chfTdp = 0, where c is an arbitrary constant. Derive hence the differential equation of the second order, equivalent (through the constant c) to one of the third order, c-'dp = VMvdUdp. CHAP. IX.] EXAMPLES TO CHAPTER IX. 245 Geodetics are, therefore, that limiting case of Didonias for which the constant c is infinite. On a plane, the Didonia is a circle, of which the equation, obtained by integration from the general form, is -or being vector of centre, and c being radius of circle. {m,) Operating by 8. Udpy the general differential equation of the Didonia takes easily the following forms : c-^Tdp =S{Uvdp.dUdp); c-^Tdp^ =:8{Ui;dp.d'p); C-' Tdp' = 8. Uvdpd^p ; , d^pdp-^ = 8 Uvdp {n.) The vector ©, of the centre of the osculating circle to a curve in space, of which the element Tdp is con- stant, has for expression, dp^ « = '' + -^- Hence for the general Didonia, Uvdp T{p--<^) = c8U^. ^ ^ vdp {o,) Hence, the radius of curvature of any one Didonia varies, in general, proportionally to the cosine of the inclination of the osculating plane of the curve to the tangent plane of the surface. And hence, by Meusnier's theorem, the difference of the squares of the curvatures of curve and surface is constant; the curvature of the surface meaning 246 QUATERNIONS. [chap. IX. here the reciprocal of the radius of the sphere which oscuhites in the redaction of the element of the Didonia. (p.) In general, for any curve on any surface, if £ denote the vector of the intersection of the axis of the element (or the axis of the circle osculating to the curve) with the tangent plane to the surface, then ^ " ''■^ S.vdpd'p ' Hence, for the general Didonia, with the same signi- fication of the symbols, $ = p—cUvdp; and the constant c expresses the length of the in- terval p— f, intercepted on the tangent phine, be- tween the point of the curve and the axis of the osculating circle. (q,) If, then, a sphere be described, which shall have its centre on the tangent plane, and shall contain the osculating circle, the radius of this sphere shall always be equal to e, (r.) The recent expression for (, combined with the first form of the general differential equation of the Didonia, gives d( = -cFdUvUdp ; Vvd^ = 0. («.) Hence, or from the geometrical signification of the constant <?, the known property may be proved, that if a developable surface be circumscribed about the arbitrary surface, so as to touch it along a Didonia, and if this developable be then unfolded into a plane, the curve will at the same time be flattened (generally) into a circular arc, with radius = c. 24. Find the condition that the equation CHAP. IX.] EXAMPLES TO CHAPTER IX. 247 may give three real values oi f for any given value of p. lif be a fiinction of a scalar parameter £, show how to find the form of this function in order that we may have „.> d'^ . d^e d'i ^ ^~ dx' ^ dy ^ dz* ~ ' Prove that the following is the relation between^ and f, ■rr-f ^^ ^{^^ in the notation of § 147. 25. Show, after Hamilton, that the proof of Dupin's theorem, that ^^ each member of one of three series of orthogonal surfaces cuts each of the other series along its lines of curvature, '^ may be expressed in quaternion notation as follows : If Svdp = 0, Sv'dp = 0, 8,vv'dp^0 be integrable, and if Svv=^ 0, then Vvdp = makes S.vvdif = 0. Or, as follows. If /S'i;Vi; = 0, ^/V2/ = 0, Sv'Vv"^^, and r.wV'= 0, then S,v'\8vV.v) = 0, 1 ' T-, ' d , d , d where v = t-=- + 9 -t~ +^-7- • dx "^ dy dz 26. Show that the equation Vap = pY^p represents the line of intersection of a cylinder and cone, of the second order, which have )3 as a common generating line. CHAPTER X. KINEMATICS. 335. \\7 *^cn a point's vector, p, is a Ainction of the time ▼ f ^, we have seen (§ 35) that its vector- velocity is expressed by --^ or, in Newton^s notation, by p. That is, if p = <t>t be the equation of an orbit, containing (as the reader may see) not merely the form of the orbit, but the law of its description also, then p = tl/t gives at once the form of the Hodograph and the law of its description. This shows immediately that the vector^acceleration (fapoinfs motion ^«p is the vector-velocity in the hodograph. Thus the fimdamental properties of the hodograph are proved almost intuitively. 336. Changing the independent variable, we have dp ds , if we employ the dash, as before, to denote -j- • This merely shows, in another form, that p expresses the velocity in magnitude and direction. But a second differentia- tion gives p = vp'\-v^p\ This shows that the vector-acceleration can be resolved into two components, the first, vp, being in the direction of motion and SECT. 338.] KINEMATICS. .249 equal in magnitade to the acceleration of the velocity^ v or -^', the second, v^p'y being in the direction of the radius of absolute curvature, and its amount equal to the square of the velocity multiplied by the curvature. [It is scarcely conceivable that this important fundamental proposition, of which no simple analytical proof seems to have been obtained by Cartesian methods, can be proved more ele- gantly than by the process just given,] 337. If the motion be in a plane curve, we may write the equation as follows, so as to introduce the usual polar coor- dinates, T and 6, 20 P = ^a'/3, where o is a imit vector perpendicular to, j8 a unit vector in, the plane of the curve. Here, of course, r and 6 wrj be considered as connected by one scalar equation,* or better, each may be looked on as a function of t. By differentiation we get which shows at once that r is the velocity along, rO that per- pendicular to, the radius vector. Again, 3£ . . ,.2(9 which gives, by inspection, the components of acceleration along, and perpendicular to, the radius vector. 338. For imiform acceleration in a constant direction, we have at once, p =z a. Whence p = d^+ft where j8 is the vector-velocity at epoch. This shows that the hodogiaph is a straight line described uniformly. Kk 260 QUAT8RKI0KS. [CHAP. X. Also p = ~^pt, no constant being added if the origin be assumed to be the position of the moving point at epoch. Since the resolved parts of p, parallel to p and o, vary re- spectively as the first and second powers of t, the curve is evi- dently a parabola (§31 (/)). But we may easily deduce from the equation the following result, the equation of a paraboloid of revolution, whose axis is a. Also S.afip = 0, and therefore the distance of any point in the path from the point ~|)3a-^j3 is equal to its distance from the line whose equation is p = — ^ a"* +«a Fap. Thus we recognise the focus and directrix property. 339. That the moving point may reach a point y we must have, for some real value of ty Now suppose T^y the velocity of projection, to be given = r, and, for shortness, write «r for {7)3. Then y = ^P^-vtm. 2 Since Tta =1, , we have (t;* — 8ay) ^* -f ^' = 0. The values of P are Hal if (f?«-«ay)*-ya»2V* 8SCT. 341.] KINBHATICS. 251 is positive. NoWj as TaTy is never less than Say^ it is evident that v^-'Say must always be positive if the roots are possible. Hence, when they are possible, both values of t* are .positive. Thus we have four values of t which satisfy the^ conditions, and it is easy to see that since, disregarding the signs, they are equal two and two, each pair refer to the same path, but de^ scribed in opposite directions between the origin and the extremity of y. There are therefore, if any, in general two parabolas which satisfy the conditions. The directions of projection kre (of course) given by the corresponding values of «r. 340. The envelop of aU the trajectories possible with a given velocity, evidently corresponds to {v^^SayY-Ta^Ty^ = 0, for then y is the vector of intersection of two indefinitely close paths in the same vertical plane. Now v*^Say=zTaTy is evidently the equation of a paraboloid of revolution of which the origin is the focus, the axis parallel to a, and the directrix plane at a distance ^p • la All the ordinary problems connected with parabolic motion are easily solved by means of the above formulae. Some, however, are even more easily treated by assuming a horizontal unit vector in the plane of motion, and expressing p in terms of it and a. But this must be left to the student. 34L For acceleration directed to or from a fixed point, we have, taking that point as origin, and putting P for the magnitude of the central acceleration, p = PUp. Whence, at once, Vpp = 0. Integrating, Vpp = y == a constant vector. The interpretation of this simple formula v&^^fi^st, p and p are X k (z 262 QUATEBKIONS. [CHAP. X- in a plane perpendicular to y, hence the path is in a plane (of course passing through the origin); second, the area of the triangle, two of whose sides are p and p, is constant. [It is scarcely possible to imagine that a more simple proof than this can be given of the fondamental &ct8, that a central orbit is a plane curve, and that equal areas are described by the radius vector in equal times.] 842. When the law of acceleraium to or from tie origin is Hat of the inv^se square of tie distance, we have where fx is negaOoe if the acceleration be directed to the origin. Hence p = ^. The following beautiful method of integration is due to Hamilton. (See Chapter lY.) a^a^Wrr ^^^ — Up .Vpp _ Up.y therefore dUp and py =:€— fiCTp, where € is a constant vector, perpendicular to y, because Syp = 0. Hence, in this case, we have for the hodograph, p = ey-^—nUp.y-^. Of the two parts of this expression, which are both vectors, the first is constant, and the second is constant in length. Hence the locus of the extremity of p is a circle, whose radius is ^, and whose centre is at the extremity of the vector ey-K ^ [This equation contains the whole theory of the Circular 8BCT. S44.] KINEMATICS- 253 Hodoffraph. Its consequences are developed at length in Hamil- ton's ElemenU."] 343. We may write the equations of this circle in the form (a sphere), and 8yp = 0, (a plane through the origin^ and through the centre of the sphere). The equation of the orbit is found by operating by V.p upon that to the hodograph. We thus obtain or y« = 8€p-\-yLTp, or iyi2}) = ^€(y^€-»-p); in which last form we at once recognise the focus and directrix property. This is in fact the equation of a conicoid of revolu- tion about its principal axis {€)y and the origin is one of the foci. The orbit is found by combining it with the equation of its plane, Syp = 0. We see at once that y*€-* is the vector distance of the directrix from the focus; and similarly that the excentricity is — , and the maior axis , , • 344. To take a simpler case : let the acceleration vary as the dUtancefram the orujfin. Then p = ±wV, the upper or lower sign being used according as the acceleration iajrom or to the centre. (d* \ -^^:m^)p = 0. Hence . p= as'^'+jSr-"*'; 264 QUATEBNIONS. [cBAF, X; or p= aoosmt-^-fimnrnt, where a and p are arbitrary, but constant, vectors ; and € is the base of Napier's logarithms. The first is the equation of a hyperbola of which a and fi are the directions of the asymptotes ; the second, that of an ellipse of which a and fi are semi-conjugate diameters. Since p = «» {ctf^— j8£~"^}, or = «{— asin«i^-|-)3cos«»^}, the hodograph is again a hyperbola or ellipse. But in the first case it is, if we neglect the change of dimensions indicated by the scalar factor m, conjugate to the orbit ; in the case of the ellipse it is similar and similarly situated. 346* Again, let the aeceleratian be as the inverse third power ofihe distance, we have Of course, we have, as usual, Fpp = y. Also, operating by S.p, fiSpp Spp = of which the integral is the equation of energy. Tp' * p.= C7-A Again, Spp=zJ^. P Hence Spp+p* = C, or Spp =1 Ct, no constant being added if we reckon the time from the passage through the apse, where Spp = 0. We have, therefore, by a second integration, p« = Ct* + Cr. ] (1) SBCT. 34&] KINEMATICS. 266 [To determine Cy remark that or p^p^ =zCH^-^y\ But p«p« = Cp^^fjL (by the equation of energy), = CH^ + C(r^,M, by(l). Hence, (7C = m— y'O To complete the solution, we have, by § 133, where )S is a unit vector in the plane of the orbit. But ri=_2.. P P' Hence log-^=_y/^. The elimination of t between this equation and (1) gives Tp in terms of Up, or the required equation of the path. We may remark that if d be the ordinary polar angle in the ^''^'*' I ^P nrr log-j- = euy. Hence we have and r*=i^{Ct* + C% from which the ordinary equations of Cotes^ spirals can be at once foimd. 346. To find tie conditions that a given curve may be the hodogra^h corresponding to a central orbit. If «r be its vector, gi/oen as a function of the time, fvr dt is that of the orbit; hence the requisite conditions are given by Yrafrndt = y, where y is a constant vector. 25& QUATBRNION& [cHAP. X. We may transform this into other shapes more resembling the Cartesian ones. Thus Vmfmdi = 0, and Y^fiadi'\- F«rw = 0. From the first fmdi = X'usj and therefore x Fmr = y, or the carve is plane. And X Vvir + Virv = ; or eliminating a?, y Vmr = — (FW«r)*. Now if t;' be the velocity in the hodograph, K its radius of curvature^ j/ the perpendicular on the tangent; this equation gives at once hif ^ Rp'\ which agrees with known results. 347. The equation of an epitrocAaid or AypotrocAoid, referred to the centre of the fixed circle^ is evidently p ^ ai* a'{-bi' a^ where a is a unit- vector in the phme of the curve and i another perpendicular to it. Here a> and 6>| are the angular velocities in the two circles^ and t is the time elapsed since the tracing point and the centres of the two circles were in one straight line. Hence, for the length of an arc of such a curve, # =/Tprf^ =/£»>/{a)»tf«H-2fiM»iaAcos(»— «i)^+«i'**h which is, of course, an elliptic Amotion. But when the curve becomes an epicycloid or a hypocycloid^ a)a+»id s= 0, and SECT. 348.] KINEMATICS. 257 which can be expressed in finite terms, as was first shown by Newton in the Prindpia, The hodograph is another curve of the same class, whose equation is 2»t 2uit p = i(a<ai ' o-f^ia)ii ^ a) j land the acceleration is denoted in magnitude and direction by the vector 2«e 2^ p = — aa)*i ^ a-'hfali » a. Of course the equations of the common Cycloid and Trochoid may be easily deduced from these forms by making a indefinitely great and a> indefinitely small, but the product ao) finite ; and transferring the origin to the point p = aa. 348. Let i be the normal- vector to any plane. Let 'CT and p be the vectors of any two points in a rigid plate in contact with the plane. After any small displacement of the rigid plate in its plane, let dvT and dp be the increments of tsr and p. Then Sidrsr = 0, Sidp = ; and, since jr('Bj— p) is constant, iS(«r— p)(rf«T — dp) = 0. And we may evidently assume dp = <i>i(p--r), disT = a)i('CJ — t); where of course t is the vector of some point in the plane, to a rotation a> about which the displacement is therefore equivalent. Eliminating it, we have dUj—p) which gives w, and thence r is at once found. For any other point a in the plane figure Sida- = 0, S(j)^(r){dp'^d<T) = 0. Hence dp— da = u)ii(p — a). L 1 258 QUATERNIONS. [CHAP. X. 5(<r— w)(Jttr— Ar) = 0. Henoe itr^dm = a>,i(<r— «r). From which^ at once^ «#, =r a>, = m, and dir = «t(<r— r), or this point also is displaced by a rotation a> about an axis through the extremity of r and parallel to u 349. In the case of a rigid body moving about a fixed point let «r, py a denote the vectors of any three points of the body; the fixed point being origin. Then vt^y p*, o-* are constant, and so are Svrp, Spc, and 8(nr. After any small displacement we have, for ^ and p, Smdv = 0, " Spdp^O, (1) 8mdp'\'Spdm =s 0. Now these three equations are satisfied by dor = VwuTj dp = Vap, where a is any vector whatever. But if d^ and dp are gtBen^ then Vdwdp = KFamVap = aS.apvr. Operate by S.Fisrp, and remember (1), S^mdp = S^pdm = S^.apw. Hence o = -^ = -^ (2) Now consider o-, Sadc = 0, Spda- ^—Sadp^ 8md<T = — 8ader» . d<r = Fcur satisfies them all, by (2), and we have thus the proposition that any small displacement of a rigid body about a fixed point is equivalent to a rotation, 350. To represent the rotation of a rigid body about a given axis, through a given finite angle. Let a be a unit-vector in the direction of the axis, p the «SCT. 351.] KINEMATICS. 259 vector of any point in the body with reference to a fixed point in the axis^ and the angle of rotation. Then p = a-^ Sap + a~^ Fap, = — aSap — a Fap. The rotation leaves^ of courscj the first part unafiected^ but the second evidently becomes or --aFapQOsd'\-Fap8mO. Hence p becomes />! = -^aSap — aFap cos 6 + Vap sin 6, = (cos- H-asin-)f)(cos- -asin-), 35L Hence to compound two rotations about axes which meet, we may evidently write, as the effect of an additional rotation ^ about ibe unit- vector fi, Hence p, = p' a' pa 'fi '. If the j3-rotation had been first, and then the a-rotation, we should have had ^ t ± -* ^I p\ = a'p'^pp 'a ', and the non-commutative property of quaternion multiplication shows that we have not, in general, p\ = Pj. If «j A y be radii of the umt sphere to the corners of a spherical triangle whose angles are -, ^, ^, we know that A A M t ± I y^fi^'a' = — 1. (Hamilton, Lectures, p. 267.) L 1 !2 260 QUATERNIONS. [chap. X. ♦it Hence /3'a'=-y"', _* t and we may write P* ^ y 'py' j OT, successive rotations about radii to two comers of a spherical triangle, and through angles double of those of the triangle, are equivalent to a single rotation about the radius to the third comer, and through an angle double of the exterior angle of the triangle. Thus any number of successive finite rotations may be com- pounded into a single rotation about a definite axis. 352. When the rotations are indefinitely small^ the effect of one is, by § 350, ^^ «, p^eVap, and for the two, neglecting products of small quantities, p. = p-^OFap-i-ttiFfip, $ and if> representing the angles of rotation about the unit vectors a and /3 respectively. But this is equivalent to representing a rotation through an angle T{6a + <f>p), about the unit- vector U($a'h4>P)' Now the latter is the direction, and the former the length, of the diagonal of the parallelogram whose sides are 6a and <^/3. We may write these results more simply, by putting a for $a, p for if>p, where a and p are now no longer unit- vectors, but represent by their versors the axes, and by their tensors the angles (small), of rotation. Thus pi = p-f Fapy p, = p-j-Fap+Fpp, =.p-\-F{a-\-p)p. 353. The general theorem, of which a few preceding sections illustrate special cases, is this : By a rotation, about the axis of q, through double the angle SECT. 354.] KINEMATICS. 261 of q, the quaternion r becomes the quaternion qrq-^. Its tensor and angle remain unchanged^ its plane or axis alone varies. A glance at the figure is suffi- • Q cient for the proof, if we note that of course T.grq;-^ = Tr, and therefore that we need consider the versor parts only. Let Q be r/K^'^ ^^^ pole o^ 9> Join CB. Then CB is j'r^'-^, its arc CB is evidently equal in length to that of r, JS'C"; and its plane (making the same angle with B'B that that of JS'C" does) has evidently been made to revolve about Q, the pole of q, through double the angle of q. If r be a vector, = p, then qpq~^ is the result of a rotation through double the angle of q about the axis of q. Hence, as Hamilton has expressed it, if B represent a rigid system, or assemblage of vectors, 9.^97^ is its new position after rotating through double the angle oi q about the axis of q. 354L To compound such rotations, we have r,qBq-^,r-^ = rq.B,{rq)-^, To cause rotation through an angle ^-fold the double of the angle of q we write i^Bq"^. To reverse the direction of this rotation write q-^'B^. To translate the body B without rotation, each point of it moving through the vector a, we write To produce rotation of the translated body about the same axis, and through the same angle, as before, q{a + B)r'> 262 QCATBRNIONS. [chap. X. Had we rotated first, and then translated, we should have had Tlie discrepance between these last results might perhaps be useful to those who do not believe in the Moon's rotation, but to such men quaternions are unintelligible. 855* By the definition of Homogeneous Strain, it is evident that if we take any three (non-coplanar) unit- vectors a, ^3, y in an unstrained mass, they become after the strain other vectors, not necessarily unit- vectors, a^, )9i, y^. Hence any other given vector, which of course may be thus expressed, p = xa-^yfi-^zy, becomes pi = xa^ ^ypi + zy^ , and is therefore known if Oi, J8i, y^ be given. More precisely pS.apy = aS.pyp^-pS.yap^-yS.aPp becomes Pi8.aPy = 4^p8.aPy == a^8.fiyp'\-fi^8.yap-\-yiS.aPp. Thus the properties of <^, as in Chapter V, enable us to study with great simplicity strains or displacements in a solid or liquid. For instance, to find a vector whose direction is unchanged by the strain, is to solve the equation Fpi^p = 0, or 4»p = gp, where ^ is a scalar unknown. [This vector equation is equivalent to three simple equations, and contains only three unknown quantities; viz. two for the direction of p (the tensor does not enter, or, rather, is a &ctor of each side), and the unknown gJ] We have seen that every such equation leads to a cubic in g which may be written g^—m^g^-\-myg^m = 0, where Wj, «»i, ;» are scalars depending in a known manner on SECT. 357.] KINEMATICS. S63 the constant vectors involved in <f>. This must have one real root, and may have tkree. 356, For simplicity let us assume that a, fi, y form a rect- angular system, then we may operate by 8,ay S.p, and S.y; and thus at once obtain the equation for ^, in the form Saa^-i-ff, Sap^, Say^ == (1) spa,, Spp.-^-g, spy, Sya,, Syp,, Syy.^g If the mass be rigid we must have Oj, ^i, y^ still rectangular unit- vectors, and therefore SaPt = Spa^,^ Say, = Sya,, (2) Syp, = SPy,,^ in which case (1) has obviously -fl as onfj root: the others being imaginary, except in two limiting cases in which their values are equal, and each is -|- 1 or — 1. [One simple method of obtaining the conditions (2) is to suppose q the quaternion by a rotation about whose axis and through double of its angle a, p, y are converted into Oi , yS, , yi . We have thus, by § 353, Oi = qaq-^ Sec, and equations (2) express the identities S.aqPgr^ = S.pqof^ &c. Numerous equally simple quaternion proofs will be obvious to the intelligent student.] 357. If we take Tp =z C we consider a portion of the mass initially spherical. This becomes of course an ellipsoid, in the strained state of the body. Or if we consider a portion which is spherical after the strain, 264 QIJATEKNIONS. [ghap. X. its initial form was T^p =s C, another ellipsoid. The relation between these ellipsoids is ob- vious from their equations. (See 311.) In either case the axes of the ellipsoid correspond to a rect- angular set of three diameters of the sphere (§ 257). But we must carefully separate the cases in which these corresponding lines in the two surfaces are^ and are not^ coincident. For^ in the former case there is pure strain, in the latter the strain is accompanied by rotation. Here we have at once the distinction pointed out by Stokes * and Helmholtz f between the cases of fluid motion in which there is, or is not, a velocity-potential. In ordinary fluid motion the distortion is of the nature of a pure strain, i. e. is differentially non-rotational ; while in vortex motion it is essentially accompanied by rotation. But the re- sultant of two pure strains is generally a strain accompanied by rotation. The question before us beautifiilly illustrates the properties of the quaternion linear and vector Amotion. 358. To find the quaternion Jbrmnla for a pure strain. Take a, /3, y now as unit-vectors parallel to the axes of the strain- ellipsoid, they become aa, ij3, cy. Hence p^ = <l>p = —aaSap^bpSpp—cySyp, And we have, for the criterion of a pure strain, the property of the function <f», that it is self -conjugate y i. e. 8p<f><T ^ 8<T<^p. 359. Two pure strains, in succession, generally give a strain accompanied by rotation. For if <f», i^ represent the strains, since they are pure we have SpiKT = Safpp,^ 8py^<T = Sinj/p. J * Cambridge PKU. Trans, 1845. t CrdU, vol. Iv, 1857. S^ iklso Phil. Mag. (Supplement) June 1867. SECT. 362.] KINEMATICS. 265 But for the compound strain we have Pi=XP = ^4>pj and we have not generally Spxo- = Saxp* For Sp\j/<t>(r = Sa<t>^p, ^y (1)> ^^^ V^<^ is not generally the same as (f>\lf. (See Ex. 7 to Chapter V.) 360. The simplicity of this view of the question might lead us to suppose that we may easily separate the pure strain from the rotation in any case, and exhibit the corresponding functions. But, for this purpose, it is generally necessary to solve the cubic equation of Chapter V in each particular case. When this is effected, the rest of the process presents no difficulty. 361. In general, if Pi = <^P = — aj/Sap— ^j/S/Sp— yi/Syp, the angle between any two lines, say p and o-, becomes in the altered state of the body cos- '{—8.U(l>pU<t>(T). The plane SCp = becomes (with the notation of § 144) /S^p, = = 8C(t>p = 8p(l/C. Hence the angle between the planes 8(p = 0, and Srjp = 0, which is cos-^ {—SMCUrj), becomes cos-' {-S.U(t>W(p'ri)' The locus of lines equally elongated is, of course, Ti^Vp = e, or Ti^p = eTpy a cone of the second order. 362. In the case of a Simple Shear y we have, obviously, Pi = (t>p = p-^pSap. M m 266 QUATERNIONS. [chap. X. The yectors which are unaltered in length are g^ven by Tp, = Tp, or 2SfipSap + P*S*ap = 0, which breaks up into S.ap = 0, and Sp(2/3+/3«a) = 0. The intersection of this plane with the plane of a, fi is per- pendicular to 2/3+i3'a. Let it be a-{-xp, then 5.(2/3-}-/3*a)(a+ar/3) = 0, i. e. 2ar— 1 = 0. Hence the intersection required is • For the axes of the strain^ one is of course afi, and the others are found by making T(l>Up 2l maximum and minimum. Let p = a+xp, then Pi = *f> = a+ar/3--/3, and Tp^ -~- — max. or mm.. gives ^•-^+^=0, from which the values of x are found. Also, as a verification, S.(a4-^i/3)(a4-3r,j3) = -1 4-/3«^iar,, and should be = 0. It is so, since, by the equation. Again ^{a4-(^i-l)^}{a4-(^,-l)/3}=-l+^»{^i^«-(^i+^,) + l}, which ought also to be zero. And, in fact, ^Tj+ara = 1 by the equation ; so that this also is verified. SECT. 364.] KINEMATICS. 267 363. We regret that our limits do not allow us to enter farther upon this very beautiful application. Bat it may be interesting here^ especially for the consideration of any continuous displacements of the particles of a mass^ to introduce another of the extraordinary instruments of analysis which Hamilton has invented. Part of what is now to be g^ven has been anticipated in last Chapter, but for continuity we commence afresh. If Fp^C (1) be the equation of one of a system of surfaces, and if the differ- ential of (1) be Svdp=:0, (2) V is a vector perpendicular to the surface, and its length is inversely proportional to the normal distance between two con- secutive surfaces. In fact (2) shows that v is perpendicular to dp, which is any tangent vector, thus proving the first assertion. Also, since in passing to a proximate surface we may write Svhp = bC, we see that F{p + v-'bC) = C+bC. This proves the latter assertion. It is evident from the above that if (1) be an equipotential, or an isothermal, surface, —v represents in direction and magni- tude the force at any point or thefltus of heat. And we have seen (§ 317) that if „ . d , d , d dx "^ dy dz d^ d^ d^ then V = VFp. This is due to Hamilton {Lectures on Qimtemions, p. 61 1). 364. From this it follows that the effect of the vector opera- tion V, upon any scalar function of the vector of a point, is to M m ^ 268 QUATERNIONS. [CHAP. X. produce the vector which represents in magnitude and direction the most rapid change in the value of the function. Let us next oonsider the effect of V upon a vector function as (r — i^+jri^-K We have at once and in this semi-Cartesian form it is easy to see that : — If o- represent a small vector displacement of a point situated at the extremity of the vector p (drawn from the origin) SVcr represents the consequent cubical compression of the group of points in the vicinity of that considered^ and W<r represents twice the vector axis of rotation of the same group of points. Similarly or is equivalent to total differentiation in virtue of our having passed from one end to the other of the vector o-. 365. Suppose we fix our attention upon a group of points which originally filled a small sphere about the extremity of p as centre, whose equation referred to that point is ^0) = ^ (1) After displacement p becomes p-\-fr, and, by last section, p -f- o) becomes p + o + o-— (/ScdV)©-. Hence the vector of the new sur- face which encloses the group of points (drawn from the ex- tremity of p-f (j) is o)i = « — (/Sft)V)(r (2) Hence co is a homogeneous linear and vector ftmction of ©i ; or and therefore, by (1), T<^a>i = e, the equation of the new surface, which is evidently a central surface of the second order, and therefore, of course, an ellipsoid. SECT. 366.] KINEMATICS. 269 We may solve (2) with great ease by approximation, if we remember that T<t is very small, and therefore that in the small term we may put Wi for « ; i. e. omit squares of small quan- tities; thus, 0) = «i4-(S'a)iV)o'. 366. If the small displacement of each point of a medium is in the direction of and proportional to, the attraction exerted at that point hy any system of material masses, the displacement is effected without rotation. For if ^ = C be the potential stirface, we have Sadp a com- plete differential ; i. e. in Cartesian coordinates ^dx+T^dy-^^dz is a differential of three independent variables. Hence the vector axis of rotation vanishes by the vanishing of each of its constituents, or r.Vcr = 0. Conversely, if there be no rotation, the displacements are in the direction of and proportional to, the normal vectors to a series of surfaces. For = 7,dpr,V(T =z {8dpV)(T'^VSiTdp, where, in the last term, V acts on cr alone. Now, of the two terms on the right, the first is a complete differential, since it may be written —I)^<t, and therefore the remaining term must be so. Thus, in a distorted system, there is no compression if 5V(r = 0, and no rotation if V,V<T = ; and evidently merely transference if <r = a = a constant vector, which is one case of Vo- = 0. 270 QdATERNIONa [chap. X. In the important case of <r = eVFp there is evidently no rotation^ since V<r = eV'Fp is evidently a scalar. In this case, then^ there are only transla- tion and compression^ and the latter is at each point proportional to the density of a distribution of matter^ which would give the potential Fp, For if r be such density^ we have at once 367. The moment of inertia of a body about a unit vector a as axis is evidently where p is the vector of the portion m of the mass^ and the origin of p is in the axis. Hence if we take kTa = e*, we have, as locus of the extremity of a. Me* =^2m{Fapy = MScufta (suppose), the momental ellipsoid. If 9 be the vector of the centre of inertia, <t the vector of m with respect to it, we have p = vr-\-(T; therefore Jffif = -2»*{(rai!r)»4-(^a<r)»} = -if (Faisr)* '^MSa(t>ia. Now, for principal axes, i is max., min., or max.-min, with the condition o« = — 1. Thus we have Sa^mFa^—fpio} = 0, Saa = ; therefore — i^ia+wFaw = joa = i^a (by operating by So), Hence (<^i4-** + 'BT»)a = +'BT&it!r, (1) • Proc. R,S.E, 1862-3. CHAP. X.] EXAMPLES TO CHAPTER X. 271 determines the values of a, k^ being found from the equation /St!r(<^-fii^ + tsr«)-*i!r = 1 (2) Now the normal to &(<(>4->t»-fi!r')-><r = 1, (3) at the point o- is (<^ + ^* 4- 'fs^Y * ^• But (3) passes through — tsr, by (2), and there the normal is (<^ + A« + t!r«)-*tsr, whichi by (1), is parallel to one of the required values of a. Thus we prove Thomson's theorem that the principal axes at any point are normals to the three surfaces, confocal with the momental ellipsoid , which pass through that point. EXAMPLES TO CHAPTER X. 1. Form, from kinematical principles, the equation of the cycloid j and employ it to prove the well-known elementary properties of the arc, tangent, radius of curvature, and evolute, of the curve. 2. Interpret, kinematically, the equation p^aU{^t^p\ where j3 is a given vector, and a a given scalar. Show that it represents a plane curve -, and give it in an integrated form independent of^. 3. If we write -car = fit — p, the equation in (2) becomes P — tsr = aUtff, Interpret this kinematically j and find an integral. What is the nature of the step we have taken in transform- ing from the equation of (2) to that of the present question ? 272 QUATERNIONS. [chap. X. 4. The motion of a point in a plane being given, refer it to (a.) Fixed rectangular vectors in the plane. (d.) Rectangular vectors in the plane^ revolving uniformly about a fixed point. (c.) Vectors, in the plane, revolving with different, but uniform, angular velocities. (d,) The vector radius of a fixed circle, drawn to the point of contact of a tangent from the moving point. In each case translate the result into Cartesian coordinates. 6. Any point of a line of given length, whose extremities move in fixed lines in a given plane, describes an ellipse. Show how to find the centre, and axes, of this ellipse ; and the angular velocity about the centre of the ellipse of the tracing point when the describing line rotates uniformly. Transform this construction so as to show that the ellipse is a hypotrochoid. 6. A point, Af moves uniformly round one circular section of a cone ; find the angular velocity of the point, a, in which the generating line passing through A meets a subcontrary section, about the centre of that section. 7. Solve, generally, the problem of finding the path by which a point will pass in the least time from one given point to another, the velocity at the point of space whose vector is p being expressed by the given scalar function Take also the following particular cases : — (a.) /p z= a while Sap> 1, fp = b while Sap< 1. {6.) fp = Sap. (c.) fp^-pK CHAP. X.] EXAMPLES TO CHAPTER X. 273 8. If, in the preceding question, fy be such a ftmction of Tp that any one swiftest path is a circle, every other such path is a circle, and all paths diverging from one point converge accurately in another. (Maxwell, Cam. and Bub, Math. Jov/rnaly IX. p. 9.) 9. Interpret, as results of the composition of successive conical rotations, the apparent truisms and '^ ^ ^ -^^=1- (Hamilton, Lectures^ p. 334.) 10. Interpret, in the same way, the quaternion operators ? = (80*(«f-')*(f«-)*> -d ?=o*a)*(j)*(i)*(^)*- ii^.) 11. Knd the axis and angle of rotation by which one given rectangular set of unit vectors o, ^3, y is changed into another given set a^, ^i, y,. 12. Show that, if the linear and vector operation <f> denotes rotation about the vector f, together with imiform expansion in all directions per- pendicular to it. Prove this also by forming the operator which produces the expansion without the rotation, and that producing the rotation without the expansion ; and jGinding their joint effect. 13. Express by quaternions the motion of a side of one right cone rolling uniformly upon another which is fixed, the vertices of the two being coincident. 14. Given the simultaneous angular velocities of a body about N n 274 QUATBENIONS. [CHAP. X. the principal axes throngli its centre of inertia^ jGind the position of these axes in space at any assigned instant. 15. Find the linear and vector function^ and also the quater- nion operator^ by which we may pass, in any simple crystal of the cubical system^ from the normal to one g^ven face to that to another. How can we use them to distinguish a series of fiices belonging to the same zone ? 16. Classify the simple forms of the cubical system by the properties of the linear and vector fiinction, or of the quaternion operator. 17. Find the vector normal of a face which truncates sym- metrically the edge formed by the intersection of two given 18. Find the normals of a pair of faces symmetrically trun- cating the given edge. 19. Find the normal of a face which is equally inclined to three given faces. 20. Show that the rhombic dodecahedron may be derived from the cube, or from the octahedron, by truncation of the edges. 21. Find the form whose faces replace, symmetrically, the edges of the rhombic dodecahedron. 22. Show how the two kinds of hemihedral forms are indi- cated by the quaternion expressions. 23. Show that the cube may be produced by truncating the edges of the regular tetrahedron. 24. Point out the modifications in the auxiliary vector function required in passing to the pyramidal and prismatic systems respectively. 25. In the rhombohedral system the auxiliary quaternion CHAP. X.] EXAMPLES TO CHAPTER X, 275 operator assumes a singularly simple form. Give this form, and point out the results indicated by it. 26. Show that if the hodograph be a circle, and the accelera- tion be directed to a fixed point; the orbit must be a conic section, which is limited to being a circle if the acceleration follow any other law than that of gravity. 27. In the hodograph corresponding to acceleration f{I)) di- rected towards a fixed centre, the curvature is inversely as 28. If two circular hodographs, having a common chord, which passes through, or tends towards, a common centre of force, be cut by any two common orthogonals, the sum of the two times of hodographically describing the two intercepted arcs (small or large) will be the same for the two hodographs. (Hamilton, EleTnents, p. 725.) 29. Employ the last theorem to prove, after Lambert, that the time of describing any arc of an elliptic orbit may be ex- pressed in terms of the chord of the arc and the extreme radii vectores. N n a CHAPTEK XL PHYSICAL APPLICATIONS. ' 868. %j^ Jfi propose to conclade the work by giving a few ▼ ▼ instances of the ready applicability of quater- nions to questions of mathematical physics^ upon which^ even more than on the Geometrical or Kinematical applications^ the real usefulness of the Calculus must mainly depend— except^ of course^ in the eyes of that. section of mathematicians for whom Transversals and Anharmonic Pencils^ &c. have a to us incompre- hensible charm. Of course we cannot attempt to give examples in all branches of physics, nor even to carry very far our in- vestigations in any one branch : this Chapter is not intended to teach Physics, but merely to show by a few examples how expressly and naturally quaternions seem to be fitted for attack- ing the problems it presents. We commence with a few general theorems in Dynamics — the formation of the equations of equilibrium and motion of a rigid system, some properties of the central axis, and the motion of a solid about its centre of inertia. 369« When any forces act on a rigid body, the force ft at the point whose vector is a, &c., then, if the body be slightly displaced, so that a becomes a -h ia, the whole work done is This must vanish if the forces are such as to maintain equili- brium. Hence tie condition ofequiliirium of a rigid body is iSfiba = 0. SECT. 371.] PHYSICAL APPLICATIONS. 277 For a displacement of translation ba is any constant vector^ hence 2)3 = (1) For a rotation-displacement, we have by § 349, € being the axis, and 2V being indefinitely small, ba = Fea, and 28.fir€a = I8.€rafi = 8.€2{rafi) = 0, whatever be 6, hence S.raj3 = (2) These equations, (1) and (2), are equivalent to the ordinary six equations of equilibrium. 370. In general, for any set of forces, let S.ro)3=a„ it is required to find the points for which the couple o^ has its axis coincident with the resultant /orce fi^. Let y be the vector of such a point. Then for it the axis of the couple is S.r(a-y)j3 = a,-ry^„ and by condition Operate by Sfi^ ; therefore and TyiSi = a,-Pr'Sa,p^ = -fi^Va^fir', or y = f^a,pr'+y?u a straight line (the Central Axis) pstrallel to the resultant force. STL To find the points about which the couple is least. Here T{ai — Vyfii) = minimum. Therefore S.{a,''Fyfi,)Vp,y = 0, 278 QUATERNIONS. [chap. XL where y' is any vector whatever. It is useless to try /= /3i^ but we may put it in succession equal to a^ and Faifii. Thus and {Va,p,y-fi]S.yra,p^ = 0. Hence y = xFa^p^ +yp^, and by operating with S.Faifii, we get or y = Fa,pr'+yfii, the same locus as in last section. 372. The couple vanishes if This necessitates 8a^ fii = 0^ or the force must be in the plane of the couple. If this be the casCj still the central asds. 373. To assign the values of forces ^, ^i, to act at e^ €i, and be equivalent to the given system. Hence r€$+ Fe, QSi -0 = ^u and f = (€-€0-^(a,-r€ii30-»-^(^~O- Similarly for ^i. The indefinite terms may be omitted^ as they must evidently be equal and opposite. In fact they are any equal and opposite forces whatever acting in the line joining the given points. 374. For the motion of a rigid system^ we have of course by the general equation of Lagrange. SECT. 376.] PHYSICAL APPLICATIONS. 279 Suppose the displacements da to correspond to a mere i^ansla- tiony then ha is any constant vector^ hence 2(m— i3) = 0, or, if Oi be the vector of the centre of inertia, and therefore we have at once and the centre of inertia moves as if the whole mass were con- centrated in it^ and acted upon hj all the applied forces. 375. Again, let the displacements ha correspond to a rotation about an axis €, passing through the origin, then ha = Fea, it being assumed that Te is indefinitely small. Hence 'S,8.€Va{ma—^) = 0, for all values of 6, and therefore 2.ra(^a-i3) = 0, which contains the three remaining ordinary equations of motion. Transfer the origin to the centre of inertia, i. e. put o=ai +«r, then our equation becomes 2r(oi-|-«r)(«wii+«2iT— jS) = 0. Or, since "S,in^ = 0, 2r«r(;»tBr— )8)+rai(ai2«»— 2)3) = 0. But ai2«»— 2)3 = 0, hence our equation is simply 2risr(»^--)3) = 0. Now 2FW)3 is the couple, about the centre of inertia, pro- duced by the applied forces ; call it 0, then ^mFfff^ = (f) (1) 376. Integrating once, ImFtff^ = y-\/<l)dt (2) 280 QUATERNIONS. [chap. XL Agaiiij as the motion considered is relative to the centre of inertia^ it most be of the nature of rotation about some axis^ in general variable. Let c denote at once the direction of^ and the angular velocity about^ this axis. Then^ evidently^ %r = Few. HencCj the last equation may be written Operating by S.€, we get 2«i(rc«r)« = ^cy-hSc/^ (3) But^ by operating directly by 2/8€dt upon the equation (1)^ we get 2«i(rw)«=-A« + 2/S€<^^. (4) (2) and (4) contain the usual four integrals of the first order. 877. When no forces act on the body^ we have ^ = 0^ and therefore 2l»«rFc«r = y, (5) 2»cr« = 2«l(r€«r)« = -**, (6) and^ from (5) and (6)^ % = -*« (7) One interpretation of (6) is^ that the kinetic energy of rota- tion remains unchanged: another is^ that the vector e ter- minates in an ellipsoid whose centre is the origin^ and which therefore assigns the angular velocity when the direction of the axis is given ; (7) shows that the extremity of the instantaneous axis is always in a plane fixed in space. Also, by (5)^ (7) is the equation of the tangent plane to (6) at the extremity of the vector €. Hence the ellipsoid (6) rolh on the plane (7). From (5) and (6)^ we have at once^ as an equation which e must satisfy^ SECT. 378.] PHYSICAL APPLICATIONS. 281 This belongs to a cone of the second degree fixed in the body. Thus all the ordinary results regarding the motion of a rigid body under the action of no forces, the centre of inertia being fixed, are deduced almost intuitively : and the only difficulties to be met with in more complex cases of such motion are those of integration, which are inherent to the subject, and appear whatever analytical method is employed. (Hamilton, Proc. KLA. 1848.) 378. We next take Presnel's Theory of Double Refraction, bnt merely for the purpose of showing how quaternions simplify the processes required, and in no way to discuss the plausibility of the physical assumptions. Let i/GT be the vector displacement of a portion of the ether, with the condition ^' =-1, (1) the force of restitution, on PresnePs assumption, is t{aH8iwr-\-6^j8jm-\-c^iSkxsr) = ^^tsr, using the notation of Chapter V. Here the function is obviously self-conjugate. «% i% c^ are optical constants de- pending on the crystalline medium, and on the colour of the light, and may be considered as given. Fresnel's second assumption is that the ether is incompres- sible, or that vibrations normal to a wave front are inadmissible. If, then, a be the unit normal to a plane wave in the crystal, we have of course a'=-l, (2) and Sanr = ; (3) but, and in addition, we have 'BT~^V^(f)'Gr II a, or Aa-BT*^ = (4) This equation (4) is the embodiment of Fresnel's second assump- tion, but it may evidently be read as meaning, tke normal to the front y the direction of vibration, and that of the force of restitution are in one plane ^ o o 282 QUATERNIONS. [chap. XI. 379. Eqaations (3) and (4)^ if satisfied by m, are also satisfied by cra^ so tbat tbe plane (3) intersects the cone (4) in two lines at rigbt angles to each other. That is, /or any given wave front there are two directions ofvidration^ and thetf are perpendicular to each other, 380. The square of the normal velocity of propagation of a plane wave is proportional to the ratio of the resolved part of the force of restitution in the direction of vibration, to the amount of displacement, hence Hence PresnePs Wave-mrface is the envelop of the plane Sap = wSm^, (5) with the conditions ot«= — 1, (1) a»=-l, (2) Sam^O, (3) S,at!T(t>w = (4) Formidable as this problem appears, it is easy enough. From (3) and (4) we get at once, xxff = F. a VaxftnT, Hence, operating by S.rff, — ar = — iScr^flj = — y*. Therefore (<^ -|- t?*)«r = — a8a(f>'Gr, and i8'.a(0+t7«)-»o= (6) In passing, we may remark that this equation gives the normal velocities of the two rays whose fronts are perpendicular to a. In Cartesian coordinates it is the well-known equation l^ m} n^ 1- H = 0. By this elimination of «r, our equations are reduced to AS.a(0 + i?^)-»a = 0, (6) v=:^Sap, (5) a^=-l (2) SECT. 381.] PHYSICAL APPLICATIONS. 283 They give at once, by § 309, (<^ + v2)-ia+«;p/Sa(0 + t;»)-»a = ka. Operating by S.a we have Substituting for k, and remarking that because <^ is self-conjugate, we have This gives at once, by rearrangement. Hence (0— p2)-ip = """^ . Operating by 5.p on this equation we have «p(0-p^)-p=-l, (7) which is the require^ equation. [It will be a good exercise for the student to translate the last ten formulae into Cartesian coordinates. He will thus reproduce almost exactly the steps by which Archibald Smith ^ first arrived at a simple and symmetrical mode of efiecting the elimination. Yet, as we shall presently see, the above process is far from being the shortest and easiest to which quaternions conduct us.] SSL The Cartesian form of the equation (7) is not the usual one. It is, of course, x'^ y^ z"^ But write (7) in the form or S'P-T^P = 0> * Cambridge Phil. Trans. 1843. O o il 284 QUATERNIONS. [chap. XI. and we have the usual expression a*ap* J«y« c*z^ Tliis last quaternion equation can also be put into either of the new forms ^ ± or y(p-«-<(>-i)-ip = 0. 382. By applying the results of §§ 171, 172 we may in- troduce a multitude of new forms. We must confine ourselves to the most simple; but the student may easily investigate others by a process precisely similar to that which follows. Writing the equation of the wave as Sp{(l>-'-^ff)-'P = 0, where we have ff = — p-% we see that it may be changed to if mSp(l>p = ffAp* =— A. Thus the new form is Sp{<l>-^-mSp(t>p)-'p = 0* (1) Here m = ^ ^ , 5/w^p = a'^ar' + iy + c«^% and the equation of the wave in Cartesian coordinates is, putting 383. By means of equation (1) of last section we may easily prove Pliicker^s Theorem. Tke Wave-Surface is its own reciprocal with respect to the ellipsoid whose equation is SECT.-S84.] PHYSICAL APPLICATIONS. 285 The equation of the plane of contact of tangents to this surface from the point whose vector is p is 8xn<tt*p = — ^ — • The reciprocal of this plane^ with respect to the unit-sphere about the origin, has therefore a vector o- where Hence p = —. — ^""^or, and when this is substituted in the equation of the wave we have for the reciprocal (with respect to the unit-sphere) of the reciprocal of the wave with respect to the above ellipsoid. 8.<r{^— J- 5<r^-i<r) or = 0, This differs from the equation (1) of last section solely in having <^-* instead of ^, and (consistently with this) — instead of m. Hence it represents the index-surface. The required reciprocal of the wave with reference to the ellipsoid is therefore the wave itself. 384. Hamilton has given a remarkably simple investigation of the form of the equation of the wave-surface, in his Elements p. 736, which the reader may consult with advantage. The following is essentially the same, but several steps of the process, which a skilled analyst would not require to write down, are retained for the benefit of the learner. Let Slip =-1 (1) be the equation of any tangent plane to the wave, i. e. of any wave-front. Then p. is the vector of wave-slowness, and the normal velocity of propagation is therefore -jjr- . Hence, if -bj be the vector direction of displacement, pr^xa \s the effective com- 286 QUATERNIONS. [cHAP. XI. ponent of the force of Testitution. Hence, ^o- denoting the whole force of restitution, we have <^ — fA-»tJ II IJL, or V II {<(>-M"')"'M. and, as «r is in the plane of the wave-front, Sfl'ST = 0, or 5/A(<^-.fi-«)-> = (2) This is, in reality, equation (6) of § 380. It appears here, however, as the equation of the Index-Surface, the polar reci- procal of the wave with respect to a unit-sphere about the origin. Of course the optical part of the problem is now solved, all that remains being the geometrical process of § 311. 385. Equation (2) of last section may be at once transformed, by the process of § 381, into Let us employ an auxiliary vector whence ft = 0*'— 0"*)^ (1) The equation now becomes *tr = 1, (2) or, by(l), fi«r«~5T0-*T = 1 (3) Differentiating (3), subtract its half from the result obtained by operating with S,t on the differential of (1). The remainder is T^S^fl — STdfJ, =r 0. But we have also (§ 311) SpdfA = 0, and therefore xp = [it^^t, where a? is a scalar. This equation, with (2), shows that Srp = (4) SECT. 387.] PHYSICAL APPLICATIONS. 287 Hence, operating on it by S.p, we have by (1) of last section and therefore p"" * = — ^i + t" * . This gives p-* = ft^— t"*. Substituting from these equations in (1) above, it becomes T-'^p-' = (p-« + T-«-(/>-Or, or T = (<^"''— P"*)~V~*- Finally, we have for the required equation, by (4), 8p-'{(l>-^—p-^)-^p-' = 0, or, by a transformation already employed, Sp{4^^p^r'p=:- 1. 386. It may assist the student in the practice of quaternion analysis, which is our main object, if we give a few of these investigations by a somewhat varied process. Thus, in § 378, let us write as in § 168, a^iSivf-Ji-b^jSjfff + c^kSkm = \'8f/vr+f/8\f'ST''jf/tir. We have, by the same processes as in § 378, 8.vray8i/isT-{-8.'UTapf8K'isr = 0. This may be written, so Jhr as the generating lines we require are concerned, 8.maVX'mp! = = 8.'usa}^'STik, since 'ora is a vector. Or we may write 8.f/F.isT\'^a = = S.f/tffy'BTa. Equations (1) denote two cones of the second order which pass through the intersections of (3) and (4) of § 378. Hence their intersections are the directions of vibration. 387. By (1) we have 8,vrK^iffafx = 0. (1) 288 QUATERNIONS. [chap. XI. Hence vX V, a, y! are coplanar ; and, as «r is perpendicnlar to a, it is equally inclined to V\'a and VyLO, For, a L, M, A be the A projections of X', fi, a on the unit sphere, £C the great circle whose pole is A, we are to find for the projections of the values of «r on the sphere points F and P', such that if LP be produced till PQ = Zp, Q may lie on the great circle AM. Hence, evidently, ^ CP ^ PB, and UP^ = PB', which proves the proposition, since the pro- jections of FX'a and Viia on the sphere are points b and c in EC, distant by quadrants from C and B respectively. 388. Or thus, 8ma = 0, S.^F.a\'vrfi = 0, therefore xw = F.aF.aX'«r/ui', = - VX^i/'-aSaFXr^Tf/. Hence {SK'i/'-x)^ = (V + o5aA')^M«^ + (M +a/Safi')S'AV. Operate by SX, and we have = Sf/rsrT^FK'a. Hence by symmetry. or 5W 5/-B = 0, and as TFxfa - TFf/a ^wa =0 , iir= U{Urk'a±Uri/a). SECT. 391.] PHYSICAL APPLICATIONS. 289 389. The optical interpretation of the common result of the last two sections is that tie planes of polarization of the two rays whose wave^fronts wre parallel, bisect the angles contained by planes passing through the normal to the wave-front and the vectors (optic axes) A', ii\ 390. As in § 380, the normal velocity is given by V* = 8m(^ = 2 Sk'taSiirff—p'^* SKkYa [This transformation, effected by means of the value of «r in § 388, is left to the reader.] Hence, if t?i, t?,, be the velocities of the two waves whose normal is a t?;-r; = 2T.rK'aVfxa oc sinX'asin/ji'a. That is, the difference of the squa/res of the velocities of the two waves varies as the product of the sines of the angles between the normal to the wave-front and the optic axes (X', ft'). 39L We have, obviously. Hence t;« = / q: {T± S). FxTa Ff/a. The equation of the index surface, for which l}, = i, Up = a, is therefore 1 = -//)« ip {T± 8). Fk'p Ff/p. This will, of course, become the equation of the reciprocal of the index-surface, i. e. the wave-surface, if we put for the fimction ^ its reciprocal : i. e. if in the values of X', /, p^ we put - , -7 , - {or a,b,c respectively. We have then, and indeed pp 290 QUATERNIONS. [chap. XI it might have been deduced even more simply as a transfonna- tion of § 380 (7), 1 =^j>p'^:{T±S).r\priip, as another form of the equation of FresneFs wave. If we employ the i, k transformation of § 121, this may be written^ as the student may easily prove^ in the form (K«-.i«)« = 5>(i-ic)p+(rrip+rricp)». 392. We may now^ in furtherance of our object^ which is to give varied examples of quaternions^ not complete treatment of any one subject^ proceed to deduce some of the properties of the wave-surface from the different forms of its equation which we have given. 393. FremeVa comtruction of tie wave by points. From § 273 (4) we see at once that the lengths of the principal semidiameters of the central section of the ellipsoid SfX^'^p = 1, by the plane Sap =s 0^ are determined by the equation Aa(<^-*-p-»)-'a = 0. If these lengths be laid off along a, the central perpendicular to the cutting plane^ their extremities lie on a surface for which a = Up, and Tp has values determined by the equation. Hence the equation of the locus is as in §§ 380^ 385. Of course the index-surface is derived firom the reciprocal eUipsoid Sp<l>p = 1 by the same construction. 394. Again^ in the equation 1 =-pp'+{T±s).r\prpLp, SBCT. 395.] PHYSICAL APPLICATIONS. 291 suppose V\p = 0, or Viip = 0, we obviously have p= ±-— or p=: ±--p> Vp vp and there are therefore four singular points. To find the nature of the sur£EU3e near these points put UK where T'gt is very small^ and reject terms above the first order in T'ur, The equation of the wave becomes, in the neighbourhood of the singular point, which belongs to a cone of the second order. 395. From the similarity of its equation to that of the wave, it is obvious that the index-surface also has four conical cusps. As an infinite number of tangent planes can be drawn at such a point, the reciprocal surface must be capable of being touched by a plane at an infinite number of points ; so that the wave- surface has four tangent planes which touch it along ridges. To find their form, let us employ the last form of equation of the wave in § 39 1 . If we put TTip = TFkp, (1) we have the equation of a cone of the second degree. It meets the wave at its intersections with the planes 5(i-K)p=± (««-.>) (2) Now the wave-surface is Umched by these planes, because we cannot have the quantity on the first side of this equation greater in absolute magnitude than that on the second, so long as p satisfies the equation of the wave. That the curves of contact are circles appears at once from (1) and (2), for they give in combination p» = +5(t+K)p, (3) Pp 5& 292 Q0ATRRNIONS. [chap. XI. the eqofttioiiB of two spheres on which the carves in question are situated. The diameter of this drcoLir ridge is TF.(i+^)U{i-K) = ^^j = l^/(a.-4.)(«•-c«). [Simple as these processes are^ the student will find on trial that the equation gives the results quite as simply. For we have only to examine the cases in which — p'* has the value of one of the roots of the symbolical cubic in ^"^^ In the present case Tp^6 is the only one which requires to be studied.] 896. By § 384^ we see that the auxiliary vector of the suc- ceeding section^ viz. is parallel to the direction of the force of restitution^ (fyar. Henoe^ as Hamilton has shown^ the equation of the wave^ in the form Stp=zO, (4) of § 385^ indicates that tAe direction of tie force of restitu^ tion is perpendicular to tie ray. Again^ as for any one versor of a vector of the wave there are two values of the tensor^ which are found from the equation S.Up{il>-''-'P'^)-^Up = 0, we see by § 393 that tAe lines of vibration for a given plane front are parallel to the axes of any section of the ellipsoid S.p<l>-^p = 1 made by a plane parallel to the front ; or to the tangents to the lines of curvature at a point where the tangent plane is parallel to the wave-front. 397. Again, a curve which is drawn on the wave-surface so as to touch at each point the corresponding line of vibration has SECT. 400.] PHYSICAL APPLICATIONS 293 Hence iS^pdp = 0, or 8p<l>p = C, so that such curves are the intersections of the wave with a series of ellipsoids concentric with it. 398. For curves cutting at right angles the lines of vibration we have ^^ y Fp<f>-' {<!>-' --p-^y^p II rp(4^-pT'p' Hence 8pdp = 0, or Tp = C, so that the curves in question lie on concentric spheres. They are also spherical conies, because where Tp = C the equation of the wave becomes 5.p(^-* + C-«)-v = o, the equation of a cyclic cone^ whose vertex is at the common centre of the sphere and the wave-surface^ and which cuts them in their curve of intersection. 399* As a final example we take the case of the action of electric currents on one another or on magnets; and the mutual action of permanent magnets. A comparison between the processes we employ and those of Ampere {Theorie des Phenomenes Electrodynamiques, 8fc,, many of which are well given by Murphy in his Electricity) will at once show how much is gained in simplicity and directness by the use of quaternions. The same gain in simplicity will be noticed in the investiga- tions of the mutual effects of permanent magnets^ where the resultant forces and couples are at once introduced in their most natural and direct forms. 400. Ampere's experimental laws may be stated as follows : I. Equal and opposite currents in the same conductor produce equal and opposite effects on other conductors : whence it follows 294 QCJATBBNIONa [cHAP. XI. that an dement of one corrent lias no effect on an element of another which lies in the plane bisecting the former at right angles. II. The effect of a conductor bent or twisted in any manner is equivalent to that of a straight one, provided that the two are traversed by equal currents, and the former nearly coincides with the latter. III. No closed circuit can set in motion an element of a cir- cular conductor about an axis through the centre of the circle and perpendicular to its plane. rV. In similar systems traversed by equal currents the forces are equal. To these we add the assumption that the action between two elements of currents is in the straight line joining them : and two others, viz. that the effect of any element of a current on another is directly as the product of the strengths of the cur- rents, and of the lengths of the elements. 40L Let there be two closed currents whose strengths are a and a^ ; let a, a^ be elements of these, a being the vector joining their middle points. Then the effect of a on a^ must, when resolved along a^, be a complete differential with respect to a (i. e. with respect to the three independent variables in- volved in o), since the total resolved effect of the closed circuit of which a' is an element is zero by III. Also by I, II, the effect is a function of Ta, 8aa% Saui, and Sa^ai, since these are sufficient to resolve a and a^ into elements parallel and perpendicular to each other and to a. Hence the mutual effect is aa,Ua/{Ta, Sua, Saa,, Saa,), and the resolved effect parallel to a^ is aaiSUai Uaf. SECT. 402.] PHYSICAL APPLICATIONS. 295 AIso^ that action and reaction may be equal in absolute mag- nitude^ y must be symmetrical in Sojol and Saa^, Again^ o( (as differential of a) can enter onh/ to the f/rst power, and mmt appear in each term of/*. Hence / = ASaa^^ + BSaaSaa^. But, by rV, this must be independent of the dimensions of the system. Hence -4 is of —2 and 5 of —4 dimensions in Ta. Therefore -^ {ASaa^Sa'a^ + BSaaS^aa^} is a complete differential, with respect to a, if £?a = a\ Let A- ^ where C is a constant depending on the units employed, there- fore d^ = ^Saa', C Ta*' and the resolved effect The factor in brackets is evidently proportional in the ordinary notation to sin ^ sin ^ cos o)— ^ cos ^ cos ^ 402. Thus the whole force is Caa^a ^S^aa^ ^ Caa^a ^ S^aa' as we should expect, d^a being = a^. [This may easily be transformed into or 5 = f , 2 Coat Ua „ ,„ , . -. — ddi (To)*, 296 QUATBRNIONS. [cHAP. XI. which is the quaternion expression for Ampere's well-known form.] 408. The whole effect on Oi of the closed circuity of which tk is an element^ is therefore Guit f a ASaaiY 2 J 8aa^ To» ' ^ CcMi e aSaai ^ f VnxC ^ between proper limits. As the integrated part is the same at both limits^ the effect is and depends on the form of the closed circuit. 404. This vector p, which is of great importance in the whole theory of the effects of closed or indefinitely extended circuits^ corresponds to the line which is called by Ampere *'directrice de Inaction Slectrodynamique/* It has a definite value at each point of space^ independent of the existence of any other current. Consider the circuit a polygon whose sides are indefinitely small; join its ang^ar points with any assumed pointy erect at the latter^ perpendicular to the plane of each elementary triangle so formed^ a vector whose length is -^ where » is the vertical angle of the triangle and r the length of one of the containing sides ; the sum of such vectors is the '^ directrice'^ at the assumed point. 405. The mere form of the result of § 403 shows at once that if the element ax be turned about its middle point, the direction of the resultant action is confmed to the plane whose normal is fi. Suppose that the element a^ is forced to remain perpendicular to some given vector b, we have SECT. 407.] PHYSICAL APPLICATIOKS. 291 and the whole action in its plane of motion is proportional to But r.hVa^p -^a^S^b. Hence the action is evidentlj constant for all possible positions of ai; or The effect of any system of closed currents on an element of a conductor which is restricted to a given jplane is {in that plane) independent of the direction of the element, 406. Let the closed current heptane and very small. Let € (where IV = I) be its normal^ and let y be the vector of any point within it (as the centre of inertia of its area); the middle point of ai being the origin of vectors. Let o = y+p; therefore d = p', and ^ = fIS± = fl(y±M to a sufficient approximation. Now (between limits) fFpp' = 2^6, where A is the area of the closed circuit. Also generally /Fyp'Syp = HSypryp+yF.y/Fpp') = (between limits) Ay Vy€, Hence for this case 407. If> instead of one small plane closed current, there be a series of such, of equal area, disposed regularly in a tubular 298 QUATBBNIONS. [c^AP. XI. form^ let x be the distance between two consecutive currents measured along the axis of the tube; then^ putting y = x€, we have for the whole effect of such a set of currents on at 2x Va f(^ . 3y£yy\ = — — - -=^ (between proper hmits). If the axis of the tubular arrangement be a closed curve this will evidently vanish. Hence a closed solenoid exerts no influence on an element of a conductor. The same is evidently true if tie solenoid be indefinite in both directions. If the axis extend to infinity in one direction^ and yo be the vector of the other extremity^ the effect is CAaa^ Fuiyo '~2x Tyl ' and is theretore perpendicular to the element and to the line joining it with the extremity of the solenoid. It is evidently inversely as Tyl and directly as the sine of the angle contained between the ^ direction of the element and that of the line joining the latter with the extremity of the solenoid. It is also inversely as Xy and there- fore directly as the number of currents in a unit of the axis of the solenoid. 408. To find the effect of the whole circuit whose element is ai on the extremity of the solenoid^ we must change the sign of the above and put a^ = yo'; therefore the effect is __ CAaa, C Vy^y, 2x J 2Vo' ' an integral of the species considered in § 403 whose value is easily assigned in particular cases. 409* Suppose the conductor to be straight, and indefinitely extended in both directions. Let M be the vector perpendicular to it from the extremity of the canal^ and let the conductor be || t], where ^ = 21; = I. . . BHCT. 410.] PHYSICAL APPLICATIONS. 299 Therefore y« = Ad+yri (where y is a scalar), and the integral in § 407 is hVnO / y ^ z. Yy%Q The whole effect is therefore CAaay^ xh Vyfi, and is thus perpendicular to the plane passing tirouffk tie con- ductor and the extremity of the canal, and varies inversely as the distance of the latter from the conductor. This is exactly the observed effect of an indefinite straight current on a magnetic pole^ or particle of free magnetism. 410. Suppose the con- ductor to be circular, and the pole nearly in its axis. Let EPD be the con- ductor^ AB its axis^ and C the pole; BC perpen- dicular to AB, and small in comparison with AE^h the radius of the circle. Let ABhea^i, BC =: bi, AP = hijx^-hy) where Then CP ^ y ^ a^i-\-bk'~'h{jx-\'ky). And the effect on -/f 2 [ff{{A-by)i-\-a^ixy-\'a^yk} a fl I — rs — 1 where the integral extends to the whole circuit. 300 QUATBRNIONS. [CHAP. XI. 41L Suppose in particular (7 to be one pole of a small magnet or solenoid CC whose length is 2 /^ and whose middle point is at 6 and distant a from the centre of the conductor. Let L CGB = A. Then evidently a^ z= a + ^oosA, b = ^sinA. Also the effect on Cbeoomes^ if a; + d'+^' = ^% 2;y^{(*-*y)*+«i^+«iy*}{i -^-jT + Y ~^ ■'■•••^ since for the whole circuit /(Ty- = 2ir 2'"(|«)' /(Toy- = 0. If we suppose the centre of the magnet fixed, the vector axis of the couple produced by the action of the current on C is ir.{i cos A + * sin ^)f^ TtA^lsinA .r Sb* 15 A*J* Ba^b cos A . 00 j8 J\ A^ ^ 2 A' A* sin A If ^, &c. be now developed in powers of I, this at once becomes ir^'^sinA »fg 6g^cosA 15a'^'cos«A 3^» S^^sin'A 15 A*^'sin*A (a + ^co8A)^cosA • 5a?cosAxn "" a«+A» "*"2" {a^ + A^y ~ dj«~+>i» ^ tf» + A« )^ ' Putting — ^ for ^ and changing the sign of the whole to get that for pole (T, we have for the vector axis of the complete couple 4TrA»;sinA.. „ ^*(4<^'--A«)(4 — 5 sin«A) „ i -(^rT^-^{^+* (^Tk^- ""^-i SECT, 413.] PHYSICAL APPLICATIONS. 301 which is almost exactly proportional to sinA if 2ei ^ ^ and I be small. On this depends a modification of the tangent galvanometer. (Bravais — Ann. de Chimiey xxxviii. 309.) 412. As before^ the effect of an indefinite solenoid on a^ is Now suppose Oi to be an element of a small plane circuity d the vector of the centre of inertia of its area, the pole of the solenoid being origin. Let y = 5+/>, then ai = />'. The whole effect is therefore CAaa^ r 2x~'J " 2xlV Vi+~lv">'' where A^ and 6i are, for the new circuit, what A and € were for the former. Let the new circuit also belong to an indefinite solenoid, and let do be the vector joining the poles of the two solenoids. Then the mutual effect is CAA^aa^ f. b' 3bSbb\ 2xx^ J^Tb^ '^ Tb' ) _ CAAiOai bo Ubp - 2xx, {TboY '^' i^oY ' which is exactly tAe mutual effect of two magnetic poles. Two Jmite solenoids J therefore^ act on each other exactly as two mxtgnetSy and the pole of an indefinite solenoid acts as a particle of free 413. The mutual attraction of two indefinitely small plane closed circuitsji whose normals are € and €i> may evidently be 302 qUATSBKIONS. [chap. XL dedaced by twice differentiatiiig the expression -^^ for the matoal action of the poles of two indefinite solenoids^ making d6 in one differentiation || c and in the other || Ct* . Bnt it may also be calculated directly by a process which will give OS in addition the CQuple impressed on one of the circuits by the other, supposing for simplicity the first to be circular. Let A and JB be the centres of inertia of the areas of A and B, c and Ci vectors normal to their planes^ <r any vector radius of JB,AB:=zp. Then whole effect on a', by §§ 406, 403, A f 3(ff^<r)504-<r)e . "" TO^i^^'V^ + tot) -*■ —mi—v + -7^) 2)3» -r Jjg, T^- y^. But between proper limits, fVi/riSetr = -^, r.jyrSei, for generally /Van S»<t = - i ( ri7(r AW<r + T. »y F. d/r<r<TO. Hence, after a reduction or two, we find that the whole force exerted by A on the centre of inertia of the area of B This, as already observed, may be at once found by twice SECT. 414.] PHYSICAL APPLICATIONS. 308 differentiating -^=^ . In the same way the vector moment^ due to Aj about the centre of inertia of B, These expressions for the whole force of one small magnet on the centre of inertia of another^ and the couple about the latter^ seem to be the simplest that can be given. It is easy to deduce from them the ordinary forms. For instance, the whole resultant couple on the second magnet OC ^ '■> may easily be shown to coincide with that given by Ellis (Camb. Math, Journaly iv. 95), though it seems to lose in sim- plicity and capability of interpretation by such modifications. 414. The above formulae show that the whole force exerted by one small magnet M, on the centre of inertia of another m^ consists of four terms which are, in order, 1st. In the line joining the magnets, and proportional to the cosine of tieir muttud inclination. 2nd. In the same line^ and proportional to five times the product of the cosines of their respective inclinations to this Hne. 3rd. and 4th. Parallel to {-..} anc( proportional to the cosine M of the inclination of { ^to the joining line. All these forces are, in addition, inversely as the fourth power of the distance between the magnets. 304 QUATERNIONS. [cHAP. XI. For the oonples about the centre of inertia of m we have 1st. J couple whose axis is perpendicular to each magnet, and which is as the sine of their mutual inclination. 2nd. A couple whose axis is perpendicular to m and to the line joining the magnets, and whose moment is as three times the product of the sine of the inclination ofxn, and the cosine of the inclination ofTAjto the joining line. In addition these couples vaiy inversely as the third power of the distance between the magnets. [These results afford a good example of what has been called the internal nature of the methods of quaternions^ reducing^ as they do at once^ the forces and couples to others independent of any lines of reference^ other than those necessarily belonging to the system under consideration. To show their ready appli- cability^ let us take a Theorem due to Grauss.] 415. If two small magnets be at right angles to each other, the moment of rotation of the first is approximately twice as great when the axis of the second passes through the centre of the fi/rst, as when the axis of the first passes through the centre of the second. In the first case ^tl/3-^Cx; C 2C" therefore moment = •mQ;T{€€^'-Z€€^ =s -^^Tcci. In the second Cill/9-^c; therefore moment = -^j^ Tt€^. Hence the theorem. i/3' 416. Again, we may easily reproduce the results of § 413, if for the two small circuits we suppose two small magnets perpendicular to their planes to be substituted, fi is then the vector joining the middle points of these magnets, and by changing the tensors we may take 26 and 2€i as the vector lengths of the magnets. SECT. 418.] PHYSICAL APPLICATIONS. 805 Hence evidently the mutual effect which is easily reducible to as before, if smaller terms be omitted. If we operate with Kci on the two first terms of the unre- duced expression, and take the difference between this result and the same with the sign of Ci changed, we have the whole vector axis of the. couple on the magnet 2€i, which is therefore, as before, seen to be proportional to 4 , . sn.fiSfie^ 417. We might apply the foregoing formulae with great ease to other cases treated by Ampdre, De Montferrand, &c.— or to two finite circular conductors as in Weber's Dynamometer — ^but in general the only difficulty is in the integration, which even in some of the simplest cases involves elliptic functions, &c., &c. 418. Let F{y) be the potential of any system upon a unit particle at the extremity of y. ^(y) = C (1) is the equation of a level surface. Let the differential of (1) be Svdy^O, (2) then r is a vector normal to (1), and is therefore the direction of the force. But, passing to a proximate level surface, we have Svby = bC. Make by = xv^ then -xTv' = bC, R r 306 QUATEBNI0K8. [cHAP. XI. Henoe v exiwesses the force in magnitude also. (§ 363.) Now by § 406 we have for the vector force exerted by a small plane closed circuit on a particle of free magnetism the ex- pression omitting the fieustors depending on the strength of the current and the strength of magnetism of the particle. Hence the potential^ by (2) and (1), area of circuit projected perpendicular to y a solid angle subtended by circuit. The constant is omitted in the integratioD^ as the potential must evidently vanish for infinite values of Ty, By means of Ampere's idea of breaking up. a finite circuit into an indefinite number of indefinitely small ones^ it is evident that the above result may be at once extended to the case of such a finite closed circuit. 419. Quaternions give a simple method of deducing the well-known property of the Magnetic Curves, Let Ay A' be two equal magnetic poles^ whose vector distance^ 2a^ is bisected in 0, QQ' an indefinitely small magnet whose length is 2p, where p = OP, Then evidently, taking moments, F(p + a)p' _ r {p-a)p' Tlp^ay - ^ T{p-ay SECT. 421.] PHYSICAL APPLICATIONS. 307 Operate by 45. Top, or S.aF(-^)U{p + a) = + {same with -a}, i. e. SadU{p + a)= ± Sad U{p^a), 8a{U{p + a)T U{p—a)} = const., or cos AOAF ± AOA'P = const., the property referred to. 420. If the vector of any point be denoted by p — ix-^jy+kz, (1) there are many physically interesting and important transforma- tions depending upon the effects of the quaternion operator '=4*4^4 <^' on various Unctions of p. When the Amotion of p is a scalar, the effect of V is to give the vector of most rapid increase. Its effect on a vector function is indicated briefly in § 364. 42L We commence with one or two simple examples, whic£ are not only interesting, but very useful in transformations. Vp = (i +&c.)(Mr + &C.)=-3 (3) V (?»• = n{Tpf-^VTp = n{TpY-'p ; (5) and, of course, v^=-^^^; (5)' whence, v^ = _^ = -^, (6) 308 QUATEBNIONS. [cHAP. XI. and, of course, V-i-=-V^ = (6)» Also, Vp=z—3 = TpVUp+VTp.Up= TpVUp—l, .'. ^Up = -±- (7) 422. B7 the help of the above results^ of whioh (6) is especially useful (though obvious on other grounds)^ and (4) and (7) very remarkable^ we may easily find the effect of v upon more complex functions. Thus, VSap=-v(«P + &c.)=-a, (1) Vra/>=-Vrpa=-.V(^-.&ip)= 3a-a = 2a (2) Hence --Tap ^ 2a_ ^ SpFap _ 2ap^'\'3pFap ^ ap^ — SpSap . „ o^rr'^ap p*SaZp—SSapSphp Sahp SSapSpbp Hence ^.«pV — = j,- =^_-___ =->w « This is a very useAil transformation in various physical appli- cations. By (6) it can be put in the sometimes more convenient form S.ipV^^iS.aV-L. (5) And it is worthy of remark that, as inay easily be seen, — iS may be put for Fin the left-hand member of the equation. 423. We have also VF^py=V{pSyp^p8Py-{-ySfip} = ^yfi^SS^^Py=Sfiy. (1) Hence, if ^ be any linear and vector Amotion of the form (l>p=:a + :EFPpy + mp, : (2) i. e. a self-conjugate function with a constant vector added, then V<^p = 25/3y— 3 1» = scalar (3) SECT. 424.] PHYSICAL APPLICATIONS. 309 Hence^ an integral of V<r = scalar constant, is <r = <^p (4) If the constant value of Vo- contain a vector part, there will be terms of the form Fep in the expression for <r, which will then express a distortion accompanied by rotation. (§ 366.) Also, a solution of Vq = a (where q and a are quaterjiions) is q = SCp+F€p + <f>p. It may be remarked also, as of considerable importance in physical applications, that, by (1) and (2) of § 422, but we cannot here enter into details on this point. 424. It would be easy to give many more of these trans- formations, which really present no difficulty ; but it is sufficient to show the ready applicability to physical questions of one or two of those already obtained ; a property of great importance, as extensions of mathematical physics are far more valuable than mere analytical or geometrical theorems. Thus, if 0- be the vector-displacement of that point of a homo- geneous elastic solid whose vector is /», we have, p being the consequent pressure produced, Vjo-hV»<r = 0, (1) whence SbpV^a- =z^8bpVp = bp, a complete diflferential. ... (2) Also, generally, p = kSVir, and if the solid be incompressible /SV<r= (3) Thomson has shown {Canii» and Dvh. Math, Journal ^ ii. p. 62), that the forces produced by given distributions of matter, elec- tricity, magnetism, or galvanic currents, can be represented at every point by displacements of such a solid producible by ex- ternal forces. It may be usefiil to give his analysis, with some additions, in a quaternion form, to show the insight gained by the simplicity of the present method. 310 QUATERNIONS. [OHAP. XI. 425. Thus, if Sahp = h -=- ^^ "^ij write, each equal to This gives <r=:-V— , the vector-force exerted by one particle of matter or free elec- tricity on another. This value of tr evidently satisfies (2) and (3). Again, if S.hpVa = d ^rr > fitter is equal to -S.hpV^^ by (4) of §422. Here a particular case is Fap which is the vector-force exerted by an element a of a current upon a particle of magnetism at p. (§ 407.) 436. Also, by § 422 (3), _ Vap ^ ap* — SpSap and we see by §§ 406, 407 that this is the vector-force exerted by a small plane current at the origin (its plane being perpen- dicular to a) upon a magnetic particle, or pole of a solenoid, at p. This expression, being a pure vector, denotes an elementary rotation caused by the distortion of the solid, and it is evident that the above value of <r satisfies the equations (2), (3), and the distortion is therefore producible by external forces. Thus the eifect of an element of a current on a magnetic particle is ex- pressed directly by the displacement, while that of a small closed current or magnet is represented by the vector-axis of the rota- tion caused by the displacement. 427. Again, let SECT. 428.] PHYSICAL APPLICATIONS. 311 It is evident that o- satisfies (2)^ and that the right-hand side of the above equation may be written Hence a particular case is and this satisfies (3) also. Hence the corresponding displacement is producible by external forces, and Vo- is the rotation axis of the element at p, and is seen as before to represent the vector-force exerted on a particle of magnetism at p by an element a of a current at the origin. 428. It is interesting to observe that a particular value of a- in this case is as may easily be proved by substitution. Again, if Sbptr =—^-^y we have evidently o- = V -^ . Now, as ^=-j is the potential of a small magnet a, at the origin, on a particle of free magnetism at p, o- is the resultant magnetic force, and represents also a possible distortion of the elastic solid by external forces, since V<r = V*<r = 0, and thus (2) and (3) are both satisfied. 312 QUATERNIONS. MISCELLANEOUS EXAMPLES. 1. The expression denotes a vector. What vector ? 2. If two Burfftces intersect along a common line of curvature^ they meet at a constant angle. 3. By the help of the quaternion formulae of rotation^ trans- late into a new form the solution (given in § 234) of the problem of inscribing in a sphere a closed polygon the directions of whose sides are given. 4. Express^ in terms of the masses^ and geocentric vectors of the sun and moon^ the sun's disturbing force on the moon^ and expand it to terms of the second order ; pointing out the magnitudes and directions of the separate components. (Hamilton^ Lectures, p. 615.) 6. If J = r*, show that 2dq = 2dr^ = i{dr+Kqdrf^)8^^ = ^{dr4-r^drKq)Sf-^ = {drq+Kqdr)q-'{q-^Kq)-^ = {drq+Kqdr){r'^li'y' _ dr+ Uj-^drUq-'^ _ drUq+ Uq-^dr _ q-^jUgdr-hdrUq-^) " Tq{Uq-\-Uq-') ^ q{Uq-\-Uq-') " Uq-\-Uq-^ q-^{qdr-\-Trdrq'^) _ drUq+Uq-^dr _ drKf^^+q-^dr = {dr + F.Fdr^q}q-' =z {dr ^F.Fdr^q''}q-' dr j^^dr F dr j^^^dr F ^ = — H-r.r ^q- FF ^q-' q q 8^ q q 8^ ^drq-'-^^FFq-'Fdr^l +-^q-'): MISCELLANEOUS EXAMPLES. 313 and give geometrical interpretations of these varied expressions for the same quantity. {Ibid. p. 628.) 6. Derive (4) of § 92 directly from (3) of § 91. 7. Find the successive values of the continued fraction =(ii-)"«- where i and J have their quaternion significations^ and x has the values 1, 2, 3, &c. {Lectures, p. 645.) 8. If we have where c is a given quaternion^ find the successive values. For what values of c does u become constant? {Ibid. p. 652.) 9. What vector is given, in terms of two known vectors, by the relation Show that the origin lies on the circle which passes through the extremities of these three vectors. 10. What problem has its conditions stated in the following six equations, from which f , 17, C are to be determined as scalar ftmctions of x, y, z, or of p = ix+jy-^iz? V«^ = 0, V^ri = 0, V^C = 0, SV^Vri = 0, S^riVC = 0, «Vf V^ = 0, 1 „ , d , d , d where V = ^^ +j^+i-^-. Show that (with a change of origin) the general solution of these equations may be put in the form where <^ is a self-conjugate linear and vector function, and ^, 77, ( s s 314 QUATEBNIONS. are to be found respectively- from the three values oif at any point by relations similar to those in Ex. 24 to Chapter IX. (See Lam^^ Journal de Mathematiques, 1843.) 11. Hamilton^ Buhop Laui^s Premium Hxamination, 1862. (a.) If OABP be four points of space^ whereof the three first are given^ and not collinear ; if also oa = a, OB =s p,ov =z p ; and if, in the equation a a the characteristic of operation F be replaced by S, the locus of F is a plane. What plane ? (i.) In the same general equation, if jP be replaced by ^ the locus is an indefinite right line. What line ? (c.) If ^ be changed to K, the locus of p is a point. What point? (d.) If jPbe made = Uj the locus is an indefinite half-line, or ray. What ray ? (e,) If ^ be replaced by T^ the locus is a sphere. What sphere ? {/,) I{ F he changed to TF, the locus is a cylinder of revolution. What cylinder ? (^.) If JF be made TFU, the locus is a cone of revolution. What cone ? (A.) If SU be substituted for Fy the locus is one sheet of such a cone. Of what cone ? and which sheet ? (i.) If F be changed to FU, the locus is a pair of rays. Which pair ? 12. {Ibid. 1863.) (a.) The equation Spp' + a^ =z MISCELLANEOUS EXAMPLES. 315 expresses that p and p' are the vectors of two points p and p', which are conjugate with respect to the sphere p» + «» = 0; or of which one is on the polar plane of the other. ( j.) Prove by quaternions that if the right line pp', connect- ing two such points^ intersect the sphere, it is cut harmonically thereby. (<?.) If p' be a given external point, the cone of tangents drawn from it is represented by the equation, and the orthogonal cone, concentric with the sphere, ty (%')'+«V = o- (rf.) Prove and interpret the equation, T{np-^a) = T(p-na)y if Tp = Ta. (e,) Transform and interpret the equation of the ellipsoid, r(ip-hpic) = K»-t«. (/.) The equation expresses that p and p' are values of conjugate points, with respect to the same ellipsoid. (ff.) The equation of the ellipsoid may also be thus written, Svp = 1, if {k^^i^Yv = (i^Kyp + 2iSKp-\-2KSip. (A,) The last equation gives also, (jc»-4«)»y = (t»-}-JC»)p»-h2FipK. (i.) "With the same signification of v, the differential equa- tions of the ellipsoid and its reciprocal become Svdp = 0, Spdv = 0. {J,) Eliminate p between the four scalar equations. Sap = a, Spp = 6, 8yp = c, Sep = e, SB 7, 316 QUATERNIONS. 13. {Ibid. 1864.) (a.) Let AiBi, A^B^j., .A^B^ ^ "^7 graven system of posited right liaes^ the 2n points being all given; and let their vector snm^ AB =r A,B,-\-A^B^-\-,.. + AJB^ be a line which does not vanish. Then a point R, and a scalar A, can be determined^ which shall satisfy the qnatemion equation, EA^.A,B,^..,-^EA^.A^^=^ A.AB; namely by assuming any origin Oj and writing, nir^ jr OA^.A,B,-{-,,.-^OA,.A^, ^^- ^ A:Bri~TAjr^ ' z ^OA^^AiBi-k-.., (b,) For any assumed point C, let Qc = CA,.A^B,+ ...'^CA^.A^B^; then this quaternion sum may be transformed as follows, Qc = Qh^CH.AB = {A + CH).AB; and therefore its tensor is, TQc = {A'+CH')^.AB, in which AB and CH denote lengths. (c) The least value of this tensor TQc is obtained by placing the point C at H; if then a quaternion be said to be a minimum when its tensor is such, we may write so that this minimum of Q^ is a vector. (rf.) The equation TQc = c = any scalar constant > TQh MISCELLANEOUS EXAMPLES. 317 expresses that the locus of the variable point C is a spheric surface^ with its centre at the fixed point Hf and with a radius r, or CH, such that so that Hj as being thus the common centre of a series of concentric spheres, determined by the given system of right lines, may be said to be the Central Point, or simply the Centre, of that system. {e,) The equation TFQc = (?i = any scalar constant > TQh represents a right cylinder, of which the radius = (cj— A«.ZB»)* divided by ZB, and of which the axis of revolution is the line, FQc= QH^h.AB; wherefore this last right line, as being the common axis of a series of such right cylinders, may be called the Central Axis of the system. (/.) The equation SQc = ^a = any scalar constant ^' represents a plane ; and all such planes are parallel to the Central Plane, of which the equation is SQc = 0. {g.) Prove that the central axis intersects the central plane perpendicularly, in the central point of the system. (A.) "When the n given vectors A^B^, ... A^B^ are parallel, and are therefore proportional to n scalars, ii, ... *«> the scalar h and the vector Qh vanish; and the centre ^is then determined by the equation, or by the expression, ^^= *, + ... + *. ' where is again an arbitrary origin. 818 QUATERNIONS. 14. (lUd. 1860.) (a.) The normal at the end of the variable vector p, to the surface of revolution of the sixth dimension^ which is represent^ by the equation {p*--a*y = 27a»(p-.a)*, (a) or by the system of the two equations^ p*^a* = 3<»aS {p-ay = ^»o», (a') and the tangent to the meridian at that pointy are respectively parallel to the two vectors, y = 2(p— a)— ^p, and T = 2(1— 2<)(p— a)4-^V; 80 that they intersect the axis a, in points of which the vectors are, respectively, 2- 1^, and ?11Z:M!». l^t' (2-Q«-2 (i.) If d^ be in the same meridian plane as p, then ^(1— Q(4-^dp = Srrf^, and 5^ = —^. (c.) Under the same condition, 4 =!<'-'>■ (rf.) The vector of the centre of curvature of the meridian^ at the end of the vector p, is, therefore, /o^^\""' 3 V 6 a— (4-Qp (tf.) The expressions in Example 38 give V* = a«^«(l-0% T« = a«^»(l-0'(4-0; hence {(t-pY = %«^«, and dp^ = |^^^'; MISCELLANEOUS EXAMPLES. 319 the radius of curvature of the meridian is^ therefore^ and the length of an element of arc of that curve is ds =1 Tdp ^ ^Ta{-^^dt. (/.) The same expressions give thus the auxiliary scalar t is confined between the limits and 4, and we may write ^ = 2 vers ^, where ^ is a real angle^ which varies continuously from to 2 IT ; the recent expression for the element of arc becomes^ therefore, ds = ZTa.td&, and gives by integration «=6Ta(d-sin^), if the arc s be measured from the point, say F, for which p = a, and which is common to all the meri- dians; and the total peripher}" of any one such curve is = 127rTa. (ff.) The value of <r gives 4(<r^-a») = 3a«^(4-^), 16 (r(wr)» = -a*^» (4-^)^ ; if, then, we set aside the axis of revolution a, which is crossed by all the normals to the siuface (a), the siuface of centres of curvature which is touched by all those normals is represented by the equation, 4(<r«— a»)» + 27a«(ra<r)« = (b) (ii.) The point F is common to the two surfaces (a) and (b), and is a singular point on each of them, being a triple point on (a), and a double point on (b); there is also at it an infinitely sharp cusp on (b), 320 QUATERNIONS. which tends to coincide with the axis a, but a determined tangent plane to (a)^ which is perpen- dicular to that axis, and to that cusp; and the pointy say F\ of which the vector = — a, is another and an exactly similar cusp on (b)^ but does not belong to (a). (i.) Besides the three universally eaincidefU intersections of the surface (a)^ with any transversal, drawn through its triple point F, in any given direction p, there are always tAree other real intersections y of which indeed one coincides with F if the transversal be perpen- dicular to the axis^ and for which the following is a general formula : (J.) The point, say F, of which the vector is p = 2o, is a double point of (a), near which that surface has a cusp, which coincides nearly with its tangent cone at that point; and the semi-angle of this cone AUXILIARY EQUATIONS. (28p{p-a)=zaH^{Z+t), , 1.2&X (p-a) = a»^»(3-4 C25(p-a)r = aH'{\^t){^^^. 'b >V.: i»*. ».♦•